Combinatorics

What is the value of $2! \times 3!$?

How many ways can we choose one vowel from the English alphabet?

In how many ways can 2 people be seated on a bench meant for 2?

If you flip a coin and then roll a 4-sided die, how many possible outcomes are there?

How many $3$-digit numbers can be formed using only the digits $7, 8, 9$ without repetition?

If there are $4$ people and only $2$ chairs in a row, how many ways can the people sit?

A bag contains $4$ red balls and $1$ blue ball. In how many ways can $2$ balls be chosen so that one is red and one is blue?

How many positive integers less than $1000$ have no digit repeated and do not contain the digit $0$?

How many ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only odd positions?

The number of ways in which $3$ prizes can be distributed among $4$ boys, such that no boy gets more than one prize, is:

In how many ways can 2 students be chosen from 10 to represent a school if one is the main representative and the other is the substitute?

How many ways can the letters of the word $EYE$ be arranged?

How many 1-digit or 2-digit numbers can be formed using the digits $1, 2, 3$ if repetition is not allowed?

If a coin is tossed 2 times and then a 6-sided die is rolled, how many outcomes are possible?

How many $4$-digit numbers can be formed from digits $1, 2, 3, 4, 5, 6$ such that the number is divisible by $5$ and repetition is not allowed?

How many $3$-digit numbers are there such that every digit is $7$?

How many $4$-digit numbers have only odd digits and no digit is repeated?

How many $5$-digit numbers formed using digits $1, 2, 3, 4, 5$ (without repetition) are divisible by $4$?

In how many ways can $4$ letters be posted in $5$ letter boxes?

What is the number of ways to arrange the letters of the word "BANANA"?

A tree diagram shows the selection of a day of the weekend (Saturday or Sunday) followed by a time of day (Morning, Afternoon, Evening). How many outcomes are there?

How many $2$-digit numbers can be formed using $\{0, 1, 2\}$ if the first digit cannot be $0$ and repetition is allowed?

A coin is tossed. If it shows Tails, a second coin is tossed. If it shows Heads, no further action is taken. How many total outcomes are in this tree diagram?

If a tree diagram for a $3$-step process has $2$ branches at step 1, $2$ branches at step 2 for every step 1 branch, and $2$ branches at step 3 for every step 2 branch, what is the total count?

In a sequence of 3 coin tosses, how many paths in the tree diagram have more Heads than Tails?

If you toss a coin and then roll a four-sided die, how many branches in the tree diagram represent 'Heads and an even number'?

A tree diagram shows all 3-digit numbers using $\{0, 1, 2\}$ where the first digit cannot be 0. If repetition is allowed, how many branches are at the final level?

A bag has 5 balls numbered 1 to 5. Two balls are drawn without replacement. How many ways can the first ball be larger than the second ball?

A person wants to travel from $(0,0)$ to $(3,1)$ on a grid by moving only Right (R) or Up (U). How many such paths are there?

In a 3-step tree diagram, the first level has 2 branches. The second level branches into 3 only if the first branch was the first option, otherwise it branches into 2. The third level always branches into 2. How many leaves are there?

Evaluate $\frac{4!}{0!}$.

What is the units digit of $10! + 5!$?

Solve for $n$: $n! = 2 \times (n-1)!$.

What is the value of $\frac{8!}{6!}$?

Find the value of the positive integer $n$ that satisfies the equation $\frac{(n+2)! + (n+1)!}{n!} = 99$.

Solve for $x$ if $x! = (3!)!$.

What is the value of $3! + 4!$?

If $x = 1! \cdot 2! \cdot 3! \cdot 4!$, what is the remainder when $x$ is divided by 10?

Find the smallest $n$ such that $n!$ is divisible by 1000.

The value of $\frac{(n+2)! + (n+1)!}{n!}$ is equal to:

How many ways can the letters of the word "NO" be arranged?

How many 2-digit numbers can be formed using digits 8 and 9 with repetition allowed?

What is the value of $\frac{5!}{0!}$?

In how many ways can 4 different colored balls be arranged in a row?

How many 6-letter words can be formed from the letters of the word $WINDOW$?

Find the number of ways to arrange the letters of the word $ROSE$ such that the consonants are always together.

How many 4-digit numbers can be formed using the digits $2, 4, 6, 8$ without repetition?

How many arrangements of the word $INDEPENDENCE$ start with the letter P?

If $^nP_5 = 20 \times ^nP_3$, find the value of $n$.

How many arrangements of the letters of the word $HEXAGON$ can be made such that the vowels occupy the even positions?

Calculate $P(2, 1) \times P(3, 2)$.

What is the value of $\frac{P(5, 3)}{P(5, 2)}$?

How many ways can 4 runners finish in the top 2 spots (1st and 2nd)?

Find the value of $P(4, 1) + P(3, 3)$.

If $P(2n-1, n) : P(2n+1, n-1) = 22 : 7$, find $n$.

How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 if digits can be repeated?

Find the number of ways in which the letters of the word "ALGEBRA" can be arranged.

How many 4-digit numbers can be formed using digits 0, 1, 2, 3, 4 such that the number is divisible by 4 and digits are not repeated?

Find the number of ways to arrange the letters of the word 'FAILURE' so that the consonants occupy only the odd positions.

In how many ways can 4 mathematics books and 3 physics books be arranged on a shelf so that all the physics books are together?

In how many ways can we choose 2 colors from a set of 8 distinct colors?

Evaluate $C(7, 1)$.

Calculate the value of $C(12, 11)$.

How many ways can you select a group of 4 players from 6 players?

In how many ways can 2 delegates be selected from 10 countries such that no two delegates are from the same country (assuming 1 delegate per country)?

Find $n$ if $^nC_3 = 4 \times ^nC_2$.

A committee of 5 is to be formed from 6 gents and 4 ladies. In how many ways can this be done if the ladies are in majority?

How many ways can 3 cards be selected from a standard deck of 52 cards such that they are all of the same suit?

How many ways can a committee of 4 be formed from 5 men and 4 women such that it contains at least one person of each gender?

How many ways can 3 letters be selected from the word 'APPLE'?

Calculate $^{8}C_{0} + ^{8}C_{1}$.

If $^{n}C_{1} = 50$, find $n$.

Evaluate $^{10}C_{1} + ^{10}C_{0}$.

What is the value of $^{5}C_{0} \times ^{5}C_{5}$?

A box contains 2 white, 3 black and 4 red balls. In how many ways can 3 balls be drawn such that at least one is black?

Find the value of $^{50}C_{47} + ^{50}C_{48}$.

In how many ways can a committee of 3 be chosen from 5 men and 3 women so that at least one woman is included?

What is the number of ways to choose 2 red balls and 2 blue balls from a bag containing 5 red balls and 6 blue balls?

If $^{n}C_{r} = 10$ and $^{n}P_{r} = 60$, find $n$.

In how many ways can a committee of 4 be chosen from 10 people if 2 specific people refuse to serve on the same committee?

A modern smartphone uses a 4-dot pattern lock on a $2 \times 2$ grid. If each dot must be used exactly once and the order matters, how many patterns are possible?

How many ways can you select a vowel from the word 'MATH'?

How many ways can a 4-digit PIN be created using the digits 0-9 if no digits can be repeated?

A person has 4 different ties. In how many ways can they choose a different tie for each of 2 days (Monday and Tuesday)?

In a lottery, a player chooses 2 numbers from $1$ to $10$. How many different combinations are possible?

How many ways can the letters of the word $QUIZ$ be arranged if the letter $Q$ and $U$ must always stay together as 'QU'?

A light panel has 5 switches, each of which can be either 'ON' or 'OFF'. How many different settings of the panel are possible?

A polygon has 54 diagonals. How many sides does this polygon have?

How many 5-digit numbers have at least one zero?

In a group of 15 students, 10 study Algebra, 8 study Geometry, and 6 study Calculus. If 3 study all three, and every student studies at least one, how many study exactly two of these subjects?