Q: How many different ways can the letters in the word 'STATISTICS' be arranged?

$$\frac{10!}{3! \times 3! \times 2!}$$

Q: A company’s password policy requires a password to consist of 2 uppercase letters (A-Z, repetition allowed) followed by 3 digits (0-9, repetition not allowed). How many different passwords can be created?

$$26^2$$ \times 10 \times 9 \times 8 = $$26^2$$ \times 720$$

Q: How many 4-digit numbers greater than 5000 can be formed using the digits 2, 3, 5, 6, 7, and 8, if no digit is repeated?

$$3 \times 5 \times 4 \times 3 = 180$$

Q: A word is formed using all the letters of 'COMMITTEE'. How many unique arrangements are possible?

$$\frac{9!}{2! \times 2! \times 2!}$$

Q: A school has 5 different math books, 4 different science books, and 3 different history books. If all books of the same subject must be kept together on a shelf, in how many ways can the books be arranged?

$$3! \times 5! \times 4! \times 3!$$

Q: A 4-digit code is required for a lock. The code must be even and use the digits 2, 3, 4, 5, 6, and 7 without repetition. How many such codes are possible?

$$3 \times 5 \times 4 \times 3 = 180$$

Q: A group of 4 boys and 3 girls are to stand in a line for a photo. In how many ways can the 3 girls stand together?

$$5! \times 3! = 120 \times 6 = 720$$

Q: A license plate consists of 2 letters (A-Z, repetition allowed) followed by 3 digits (0-9, repetition allowed). How many unique license plates can be made?

$$26^2$$ \times $$10^3 = 676$$,000$$

Q: A 4-digit number is to be formed using the digits 1, 2, 3, 4, 5, and 6, with no repetition. How many such numbers are odd?

$$3 \times 5 \times 4 \times 3 = 180$$

Q: A group of 6 people is to be seated in a row. If two particular people must not sit together, how many possible arrangements are there?

$$6! - 5! \times 2 = 720 - 240 = 480$$

Question 1

A restaurant offers 4 types of sandwiches, 3 types of salads, and 2 types of drinks. If a customer orders one item from each category, how many different meal combinations are possible?

Accepted Answer

$$4 	imes 3 	imes 2 = 24$$

Question 2

A security system requires a 5-digit code using the digits 0-9, where digits may be repeated. How many unique codes are possible?

Accepted Answer

$$10^5 = 100$$,000$$

Question 3

How many different ways can the letters in the word 'STATISTICS' be arranged?

Accepted Answer

$$\frac{10!}{3! 	imes 3! 	imes 2!}$$

Question 4

A company’s password policy requires a password to consist of 2 uppercase letters (A-Z, repetition allowed) followed by 3 digits (0-9, repetition not allowed). How many different passwords can be created?

Accepted Answer

$$26^2$$ 	imes 10 	imes 9 	imes 8 = $$26^2$$ 	imes 720$$

Question 5

How many 4-digit numbers greater than 5000 can be formed using the digits 2, 3, 5, 6, 7, and 8, if no digit is repeated?

Accepted Answer

$$3 	imes 5 	imes 4 	imes 3 = 180$$

Question 6

A word is formed using all the letters of 'COMMITTEE'. How many unique arrangements are possible?

Accepted Answer

$$\frac{9!}{2! 	imes 2! 	imes 2!}$$

Question 7

A school has 5 different math books, 4 different science books, and 3 different history books. If all books of the same subject must be kept together on a shelf, in how many ways can the books be arranged?

Accepted Answer

$$3! 	imes 5! 	imes 4! 	imes 3!$$

Question 8

A 4-digit code is required for a lock. The code must be even and use the digits 2, 3, 4, 5, 6, and 7 without repetition. How many such codes are possible?

Accepted Answer

$$3 	imes 5 	imes 4 	imes 3 = 180$$

Question 9

In a race with 7 runners, how many different ways can the gold, silver, and bronze medals be awarded?

Accepted Answer

$$7 	imes 6 	imes 5 = 210$$

Question 10

A committee of 3 is to be formed from 8 people. If the order of selection matters (i.e., president, vice-president, secretary), how many different committees are possible?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 11

How many different 3-letter 'words' (not necessarily meaningful) can be formed from the letters A, B, C, D, and E if repetition is allowed?

Accepted Answer

$$5^3 = 125$

Question 12

A group of 4 boys and 3 girls are to stand in a line for a photo. In how many ways can the 3 girls stand together?

Accepted Answer

$$5! 	imes 3! = 120 	imes 6 = 720$$

Question 13

A license plate consists of 2 letters (A-Z, repetition allowed) followed by 3 digits (0-9, repetition allowed). How many unique license plates can be made?

Accepted Answer

$$26^2$$ 	imes $$10^3 = 676$$,000$$

Question 14

How many ways can the letters in the word 'BALLOON' be arranged?

Accepted Answer

$$\frac{7!}{2! 	imes 2!}$$

Question 15

A 4-digit number is to be formed using the digits 1, 2, 3, 4, 5, and 6, with no repetition. How many such numbers are odd?

Accepted Answer

$$3 	imes 5 	imes 4 	imes 3 = 180$$

Question 16

A group of 6 people is to be seated in a row. If two particular people must not sit together, how many possible arrangements are there?

Accepted Answer

$$6! - 5! 	imes 2 = 720 - 240 = 480$$

Question 17

A company uses a code consisting of 2 letters (A, B, C, D, E) and 2 digits (0-9, no repetition, digits cannot start with 0). How many unique codes can be created?

Accepted Answer

$$5^2$$ 	imes 9 	imes 8 = 1800$$

Question 18

How many different ways can 8 people be seated in a row if 3 specific people must sit together?

Accepted Answer

$$6! 	imes 3! = 720 	imes 6 = 4320$$

Question 19

A word is formed from all the letters of 'UNIVERSITY'. How many unique arrangements are possible?

Accepted Answer

$$\frac{10!}{2!}$$

Question 20

A group of 3 South Africans, 2 Australians, and 2 Englishmen are to be randomly assigned to 7 adjacent hotel rooms. What is the probability that the 2 Australians and the 2 Englishmen each have adjacent rooms?

Accepted Answer

$$\frac{5! 	imes 2! 	imes 2!}{7!}$$

Step-by-Step Guidance for Counting, Permutations, and Probability (Statistics for Business)

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