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

$$\frac{10!}{3! \times 3! \times 2! \times 1! \times 1!} = 50400$$

Q: A company’s client code consists of 2 letters (A–E, repetition allowed) followed by 3 digits (0–9, no repetition, cannot start with 0). How many unique client codes are possible?

$$5 \times 5 \times 9 \times 9 \times 8 = 16,200$$

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

$$4 \times 5 \times 4 \times 3 = 240$$

Q: A password consists of 3 digits (1–9, repetition allowed) followed by 2 vowels (A, E, I, O, U, no repetition). How many different passwords can be created?

$$9^3$$ \times 5 \times 4 = 2,916 \times 20 = 58,320$$

Q: How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

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

Q: A group of 6 people is to be seated in a row. In how many ways can they be seated if two particular people refuse to sit next to each other?

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

Q: How many different 4-digit numbers greater than 4000 can be formed using the digits 1, 2, 3, 4, and 5, each exactly once?

$$3 \times 4 \times 3 \times 2 = 72$$

Q: A 4-digit code is required to open a lock. The code must be even-numbered and may not contain the digits 0 or 1. Digits may not be repeated. How many possible codes are there?

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

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

$$\frac{9!}{4! \times 2!} = 7560$$

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

$$\frac{5! \times 2! \times 2!}{7!} = \frac{480}{5040} = \frac{1}{10.5}$$

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 can 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

$$\frac{10!}{3! 	imes 3! 	imes 2! 	imes 1! 	imes 1!} = 50400$$

Question 4

A company’s client code consists of 2 letters (A–E, repetition allowed) followed by 3 digits (0–9, no repetition, cannot start with 0). How many unique client codes are possible?

Accepted Answer

$$5 	imes 5 	imes 9 	imes 9 	imes 8 = 16,200$$

Question 5

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

Accepted Answer

$$4 	imes 5 	imes 4 	imes 3 = 240$$

Question 6

A password consists of 3 digits (1–9, repetition allowed) followed by 2 vowels (A, E, I, O, U, no repetition). How many different passwords can be created?

Accepted Answer

$$9^3$$ 	imes 5 	imes 4 = 2,916 	imes 20 = 58,320$$

Question 7

How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

Accepted Answer

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

Question 8

A group of 6 people is to be seated in a row. In how many ways can they be seated if two particular people refuse to sit next to each other?

Accepted Answer

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

Question 9

How many 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 10

A committee of president, vice-president, and secretary is to be formed from 8 people. How many different arrangements are possible if no person can hold more than one position?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 11

How many different 4-digit numbers greater than 4000 can be formed using the digits 1, 2, 3, 4, and 5, each exactly once?

Accepted Answer

$$3 	imes 4 	imes 3 	imes 2 = 72$$

Question 12

A word is formed by arranging all the letters of 'SCHOOL'. How many unique arrangements are possible?

Accepted Answer

$$\frac{6!}{2!} = 360$$

Question 13

A 4-digit code is required to open a lock. The code must be even-numbered and may not contain the digits 0 or 1. Digits may not be repeated. How many possible codes are there?

Accepted Answer

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

Question 14

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

Accepted Answer

$$\frac{9!}{4! 	imes 2!} = 7560$$

Question 15

A group of 3 South Africans, 2 Australians, and 2 Englishmen are 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

$$\frac{5! 	imes 2! 	imes 2!}{7!} = \frac{480}{5040} = \frac{1}{10.5}$$

Question 16

How many different 3-letter 'words' can be formed from the letters of the word 'KEYBOARD' if no letter is repeated?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 17

A group of 8 learners is to be seated in a row. In how many ways can they be seated if 3 specific learners must sit together?

Accepted Answer

$$6! 	imes 3! = 720 	imes 6 = 4,320$$

Question 18

A company uses a code consisting of two letters (A–E, repetition allowed) and two digits (0–9, no repetition). How many unique codes are possible?

Accepted Answer

$$5 	imes 5 	imes 10 	imes 9 = 2,250$$

Question 19

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

Accepted Answer

$$\frac{11!}{4! 	imes 4! 	imes 2! 	imes 1!} = 34,650$$

Question 20

A group of 10 athletes is to be assigned to 10 lanes for a race. Four athletes insist on being placed in lanes next to each other. In how many different ways can the lanes be allocated?

Accepted Answer

$$7! 	imes 4! = 5,040 	imes 24 = 120,960$$

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

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