10. Sequences, Induction, and Probability

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

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

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. (3x + y)^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. (2x + 1)^4

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. (y-3)^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. (c+2)^5

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. (3x − y)^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+2)^8

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. (x² + 1)^16

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. (2x + y)^6; third term

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. (x²+y³)^8; sixth term

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

In Exercises 39–48, find the term indicated in each expansion. (x² + y)^22; the term containing y^14

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

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

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

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

Evaluate the given binomial coefficient 11 8

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

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

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

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.

Find the term in the expansion of (x² + y²)^5 containing x^4 as a factor.

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)!r! for n = 8 and r = 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. (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

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)! for n = 20 and r = 3