Sequences and Summation Notation

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−3)^n

In Exercises 8–9, find each indicated sum. This is a summation, refer to the textbook.

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=7 and a_n=a_n-1 + 5 for n≥2

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=2(n+1)!

In Exercises 29–42, find each indicated sum. 6Σi=1 5i

In Exercises 29–42, find each indicated sum. 9Σi=5 11

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 2+2^2+2^3+⋯+ 2^11

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=4 (a_i/b_i)^2

In Exercises 81–85, use a calculator’s factorial key to evaluate each expression. 200!/198!

In Exercises 81–85, use a calculator’s factorial key to evaluate each expression. 54!/(54−3)!3!

Write the first five terms of the sequence whose first term is 9 and whose general term is an= (an−1)/2 if a_n-1 is even, a_n=3a_n-1 + 5 if a_n-1 is odd for n≥2.

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=3^n

Evaluate 40!/(4! 38!)

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)^n+1/(2^n−1)

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n = n^2/n!

In Exercises 23–28, evaluate each factorial expression. (n+2)!/n!

In Exercises 29–42, find each indicated sum. 4Σi=1 (−1/2)^i

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1^2+2^2+3^2+⋯+ 15^2

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 4+4^2/2+4^3/3+⋯+ 4^n/n

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+(a+d)+(a+2d)+⋯+ (a+nd)

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 (a_i+3b_i)

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 a_i^2−5Σi=3 b_i^2

In Exercises 81–85, use a calculator’s factorial key to evaluate each expression. 20!/(20−3)!

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=3n+2

In Exercises 1–6, write the first four terms of each sequence whose general term is given. a_n = 1/(n - 1)!

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=2n/(n+4)

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=4 and a_n=2a_n-1 + 3 for n≥2

In Exercises 23–28, evaluate each factorial expression. 16!/2!14!

In Exercises 29–42, find each indicated sum. 5Σk=1 k(k+4)

In Exercises 29–42, find each indicated sum. 5Σi=1 i!/(i−1)!

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+ar+ar2+⋯+ ar^12

In Exercises 61–68, use the graphs of and to find each indicated sum.

5∑i=1 (2a_i+b_i)

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 a_i^2+5Σi=1 b_i^2

In Exercises 81–85, use a calculator’s factorial key to evaluate each expression. 20!/300

In Exercises 1–6, write the first four terms of each sequence whose general term is given. a_n = 7n - 4

In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)^n(n+3)

In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=3 and a_n=4a_n-1 for n≥2

In Exercises 23–28, evaluate each factorial expression. 17!/15!

In Exercises 29–42, find each indicated sum. 4Σi=1 2i^2

In Exercises 29–42, find each indicated sum. 4Σi=0 (−1)^i/i!

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. 5+7+9+11+⋯+ 31

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=1 (a_i^2+1)

In Exercises 61–68, use the graphs of and to find each indicated sum.

5Σi=4 (a_i/b_i)^3

In Exercises 81–85, use a calculator’s factorial key to evaluate each expression. (300/20)!

In Exercises 1–14, write the first six terms of each arithmetic sequence. a₁ = 200, d = 20

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 +6, a₁ = −9

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a50 when a₁ = 7, d = 5.

In Exercises 19–21, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. -7, -3, 1, 5 ...

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. 7,3,-1,-5,...

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an= an-1 -10, a₁ = 30

Find the sum of the first 60 positive even integers.

Find the sum of the even integers between 21 and 45.

For Exercises 45–50, write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1= 5/2, d = -1/2

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a6 when a₁ = 13, d = 4.

In Exercises 16–18, find the indicated term of the arithmetic sequence with first term, , and common difference, d. Find a12 when a1 = -8, d = -2

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. 1, 5, 9, 13,...

In Exercises 25–27, use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. The is a summation, refer to the textbook.

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an = an-1 +3, a₁ = 4

Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.

For Exercises 45–50, write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

Use the formula a_n=4+(n-1)(-7) to find the eighth term of the sequence 4, −3, −10,…

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1 = 300, d= -90

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 -0.4, a₁ = 1.6

In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a₁ = 35, d = -3.

Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. a1=-20, d = -4

Find the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....

For Exercises 45–50, write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1= -8, d=5

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 -10, a₁ = 30

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a200 when a₁ = −40, d = 5.

Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. a1 = 9, d=2

Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,……….

Find the sum of the odd integers between 30 and 54.

In the sequence 21,700, 23,172, 24,644, 26,116,... which term is 314,628?

In Exercises 1–8, write the first five terms of each geometric sequence. a1 = 20, r = 1/2

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a8 when a1 = 1 000 000, r = 0.1

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2

In Exercises 37–44, find the sum of each infinite geometric series. 3 + 3/4 + 3/4^2 + 3/4^3 + ...

Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n + 5

In Exercises 63–64, find a2 and a3 for each geometric sequence. 2, a2, a3, - 54

In Exercises 1–8, write the first five terms of each geometric sequence. a1 = 5, r = 3

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a40 when a1 = 1000, r = - 1/2

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 7) when a(sub 1) = 2, r = 3

In Exercises 37–44, find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...

Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.257 ̅ (repeating 257)

In Exercises 63–64, find a2 and a3 for each geometric sequence. 8, a2, a3, 27

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a12 when a1 = 5, r = - 2

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 0.0004, - 0.0004, 0.04, - 0.04, ...

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 1/2, r = 1/2

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 6 Σ (i = 1) (1/2)^(i + 1)

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum. *sigma notation* i=1 start, 6 end, 5^i formula

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.47 ̅ (repeating 47) = 47/100 + 47/10,000 + 47/1,000,000 + ...

In Exercises 57–62, let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a8 when a1 = 6, r = 2

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 1.5, - 3, 6, -12, ...

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 3, r = 2

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 10 Σ (i = 1) 5 · 2^i

Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.5 ̅ (repeating 5) = 5/10 + 5/100 + 5/1000 + 5/10,000 + ...

In Exercises 57–62, let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a10 + b10.

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5^n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

In Exercises 1–8, write the first five terms of each geometric sequence. an = - 5a_(n-1), a1 = - 6

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 18, 6, 2, 2/3, ...

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 8 Σ (i = 1) 3^i

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...

In Exercises 37–44, find the sum of each infinite geometric series. ∞ Σ (i = 1) 8(- 0.3)^(i -- 1)

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n^2 + 5

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

In Exercises 1–8, write the first five terms of each geometric sequence. an = - 4a_(n-1), a1 = 10

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 3, 12, 48, 192, ...

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 5) when a(sub 1) = -3, r = 2

In Exercises 37–44, find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...

Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = 2^n

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 3 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely. Sn: 4 + 8 + 12 + ... + 4n = 2n(n + 1)

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely. Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 2 + 4 + 8 + ... + 2^n = 2^(n+1) - 2

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ... + 1/(n(n+1)) = n/(n + 1)

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Use mathematical induction to prove that the statement is true for every positive integer n. 1 + 4 + 4^2 + ... + 4^(n-1) = ((4^n)-1)/3

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 1 + 3 + 5 + ... + (2n - 1) = n^2

In Exercises 1–4, a statement S_n about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 3 is a factor of n^3 - n.

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely. Sn: 2 is a factor of n^2 - n + 2.

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n^2

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1

In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n^2 - n.

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. n Σ (i = 1) 5 · 6^i = 6(6^n - 1)

In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. (ab)^n = a^n b^n

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

In Exercises 1–8, evaluate the given binomial coefficient.

In Exercises 1–8, evaluate the given binomial coefficient.

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x + y)^3

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x + 1)^4

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)^4

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)^5

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)^5

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)^8

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)^16

In Exercises 39–48, find the term indicated in each expansion. (2x + y)^6; third term

In Exercises 39–48, find the term indicated in each expansion. (x²+y³)^8; sixth term

In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^3 +x^-2)^4

n Exercises 53–54, find f(x + h) − f(x)/h and simplify. f(x) = x^4+7

Evaluate the given binomial coefficient 11 8

Use the Binomial Theorem to expand the binomial and express the result in simplified form. ((x^2)-1)^4

Find the term indicated in the expansion. (2x-3)^6; fifth term

Use the Binomial Theorem to expand and then simplify the result: (x² +x+ 1)³.

In Exercises 1–8, evaluate the given binomial coefficient.

In Exercises 1–8, evaluate the given binomial coefficient.

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (5x – 1)^3

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)^4

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x³ − 1)^4

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)^5

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)^6

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)^10

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (y³ − 1)^20

In Exercises 39–48, find the term indicated in each expansion. (x − 1)^9; fifth term

In Exercises 39–48, find the term indicated in each expansion. (x − 1/2)^9; fourth term

In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^1/3 +x^-1/3)^3

Find the middle term in the expansion of (3/x + x/3)^10

Use the Binomial Theorem to expand the binomial and express the result in simplified form. (2x+1)^3

Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9

In Exercises 1–8, use the formula for nPr to evaluate each expression. 9P4

In Exercises 1–8, use the formula for nPr to evaluate each expression. 8P0

In Exercises 9–16, use the formula for nCr to evaluate each expression. 5C0

In Exercises 21–28, evaluate each expression.

Use the Fundamental Counting Principle to solve Exercises 29–40. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

How many four-digit odd numbers less than 6000 can be formed using the digits 2, 4, 6, 7, 8, and 9?

In Exercises 1–8, use the formula for nPr to evaluate each expression. 6P6

In Exercises 9–16, use the formula for nCr to evaluate each expression. 7C7

In Exercises 21–28, evaluate each expression.

Use the Fundamental Counting Principle to solve Exercises 29–40. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or sport). In how many ways can you order the car?

Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Evaluate the expression. *permutation notation* the number of permutations 9 things taken 5 at a time (sub 9)P(sub 5)

In how many ways can five airplanes line up for departure on a runway?

In Exercises 1–8, use the formula for nPr to evaluate each expression. 8P5

In Exercises 9–16, use the formula for nCr to evaluate each expression. 11C4

In Exercises 21–28, evaluate each expression.

In Exercises 21–28, evaluate each expression.

Use the Fundamental Counting Principle to solve Exercises 29–40. An ice cream store sells two drinks (sodas or milk shakes) in four sizes (small, medium, large, or jumbo) and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

Evaluate the expression. *permutation notation* the number of permutations 8 things taken 3 at a time (sub 8)P(sub 3)

How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

In Exercises 11–16, a die is rolled. Find the probability of getting an odd number.

In Exercises 39–44, you are dealt one card from a 52-card deck. Find the probability that you are not dealt a king.

If you toss a fair coin six times, what is the probability of getting all heads?

What is the probability of a family having five boys born in a row? (Assume the probability of a male birth is 1/2.)

In Exercises 11–16, a die is rolled. Find the probability of getting a 4.

In Exercises 21–22, a fair coin is tossed two times in succession. The sample space of equally likely outcomes is {HH,HT,TH,TT}. Find the probability of getting two heads.

In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

You are dealt one card from a 52-card deck. Find the probability of getting an ace or a king.

In Exercises 17–20, you are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card.

In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

A die is rolled. Find the probability of getting a number less than 3 or greater than 4.

The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of no flooding for four consecutive years?

In Exercises 17–20, you are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen.

In Exercises 45-46, it is equally probable that the pointer on the spinner shown will land on any one of the eight regions, numbered 1 through 8. If the pointer lands on a borderline, spin again. Find the probability that the pointer will stop on an odd number or a number less than 6.

A die is rolled. Find the probability of getting a number less than 5.

The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of a flood for three consecutive years?

In Exercises 11–16, a die is rolled. Find the probability of getting a number greater than 4.

In Exercises 39–44, you are dealt one card from a 52-card deck. Find the probability that you are not dealt a picture card.

The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of a flood two years in a row?