Q: Suppose a password consists of 3 uppercase letters followed by 2 digits. How many different passwords are possible if letters and digits can be repeated?

$$26 \times 26 \times 26 \times 10 \times 10$$

Q: A committee of 4 is to be formed from 6 men and 5 women. How many different committees can be formed if the committee must have exactly 2 men and 2 women?

$$\binom{6}{2} \times \binom{5}{2}$$

Q: A university assigns student IDs using 2 letters followed by 4 digits, with no repetition allowed. How many unique IDs are possible?

$$26 \times 25 \times 10 \times 9 \times 8 \times 7$$

Question 1

Suppose a password consists of 3 uppercase letters followed by 2 digits. How many different passwords are possible if letters and digits can be repeated?

Accepted Answer

$$26 	imes 26 	imes 26 	imes 10 	imes 10$$

Question 2

A committee of 4 is to be formed from 6 men and 5 women. How many different committees can be formed if the committee must have exactly 2 men and 2 women?

Accepted Answer

$$\binom{6}{2} 	imes \binom{5}{2}$$

Question 3

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

Accepted Answer

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

Question 4

A university assigns student IDs using 2 letters followed by 4 digits, with no repetition allowed. How many unique IDs are possible?

Accepted Answer

$$26 	imes 25 	imes 10 	imes 9 	imes 8 	imes 7$$

Question 5

In a knockout tournament with 8 players, how many different ways can the players be paired in the first round?

Accepted Answer

$$\frac{8!}{2!2!2!2!4!}$$

Question 6

How many functions $$f$$ from a set of $$n$$ elements to $$\{0,1\}$$ are possible?

Accepted Answer

$$2^n$$

Question 7

A bag contains 5 red balls and 7 blue balls. How many ways can you choose 3 balls such that at least one is red?

Accepted Answer

$$\binom{12}{3} - \binom{7}{3}$$

Question 8

How many different signals can be made by arranging 3 green, 2 yellow, and 4 red flags in a line if flags of the same color are identical?

Accepted Answer

$$\frac{9!}{3!2!4!}$$

Question 9

A class has 8 students. In how many ways can they be ranked from 1st to 8th if no two students can share a rank?

Accepted Answer

$$8!$$

Question 10

How many ways can a president, vice-president, and secretary be chosen from a club with 10 members, if no person can hold more than one office?

Accepted Answer

$$10 	imes 9 	imes 8$$

Question 11

How many different 5-card hands can be dealt from a standard deck of 52 cards?

Accepted Answer

$$\binom{52}{5}$$

Question 12

A group of 12 people is to be split into 3 teams of 4. How many ways can this be done if the teams are unlabeled?

Accepted Answer

$$\frac{12!}{(4!)^3$ 3!}$$

Question 13

How many ways can 5 identical balls be distributed into 3 distinct boxes?

Accepted Answer

$$\binom{5+3-1}{3-1}$$

Question 14

A pizza shop offers 8 toppings. How many different pizzas can be made if each pizza can have any number of toppings (including none)?

Accepted Answer

$$2^8$$

Question 15

How many ways can 4 people be seated in a row of 10 chairs?

Accepted Answer

$$\binom{10}{4} 	imes 4!$$

Combinatorial Analysis and Basic Counting Principles in Probability

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