Permutations and Combinations

How many $4$-digit numbers can be formed where no digit is repeated? (Note: The first digit cannot be $0$)

A quiz has $3$ questions. The first two are True/False, and the third is multiple-choice with $4$ options. In how many ways can the quiz be answered?

In how many ways can $2$ letters be posted in $3$ different letter boxes?

How many $2$-digit numbers can be formed with distinct digits?

In a library, there are $12$ different books on Fiction, $8$ different books on Science, and $10$ different books on History. A student wants to borrow two books such that one is from the Fiction section and the other is from either the Science section or the History section. In how many ways can the student make this selection?

A person wants to buy a car and has a choice of $5$ models, $7$ colors, and $3$ types of engines. How many different choices does he have?

How many $3$-digit numbers are there such that all their digits are odd and repetition is allowed?

How many 4-digit numbers can be formed using the digits $1, 2, 3, 4$ with repetition allowed such that the sum of the digits is an even number?

How many 3-digit numbers can be formed using the digits $0, 1, 2, 3, 4, 5$ with repetition such that at least one digit is zero?

Find the number of 4-digit numbers such that the thousand's digit is strictly greater than the hundred's digit.

Find the number of ways to arrange the letters of the word 'BIRD'.

Evaluate $\frac{6!}{3!}$.

How many ways can 4 distinct prizes be awarded to 4 students such that each student gets one prize?

Find the number of 3-digit numbers that can be formed using digits $2, 4, 6$ without repetition.

In how many ways can the letters of the word 'TRIANGLE' be arranged if the letter 'T' is fixed at the first position?

How many arrangements of the letters of the word 'MONDAY' can be formed starting with a vowel?

In how many ways can the letters of the word 'PROBABILITY' be arranged so that the arrangements begin with 'P' and end with 'Y'?

The number of ways in which the letters of the word 'ENGINEERING' can be arranged is:

In how many ways can 6 Indians and 6 Americans sit around a round table so that no two Indians sit together?

How many 5-digit numbers can be formed using the digits $1, 2, 3, 4, 5$ without repetition which are divisible by 4?

Find the value of $^{8}C_{7}$.

In how many ways can you choose zero items from a set of 10 distinct items?

Evaluate the value of $^{9}C_{1}$.

In how many ways can 2 distinct pens be chosen from a set of 4 different pens?

A committee of $5$ members is to be formed from a group of $6$ men and $5$ women. In how many ways can this be done such that the committee contains more women than men?

There are $10$ points in a plane, and no three points are collinear. How many distinct triangles can be formed by joining these points?

If $^{n}C_{3} + ^{n}C_{2} = ^{6}C_{3}$, then the value of $n$ is:

In how many ways can $5$ different rings be worn on $4$ fingers?

How many ways can $3$ girls and $3$ boys be seated in a row so that no two girls are together?

The number of ways in which $10$ identical coins can be put into $3$ distinct pockets such that no pocket is empty is:

Simplify the fraction $\frac{20!}{19!}$.

Calculate $4! + 0!$.

If $n! = 120$, what is the value of $(n-1)!$?

Find the value of $\frac{5! + 4!}{4!}$.

Find the value of $n$ that satisfies the equation $\frac{(n+2)! + (n+1)!}{(n+1)! - n!} = \frac{48}{5}$.

If $\frac{1}{7!} + \frac{1}{8!} = \frac{x}{9!}$, find $x$.

Find the value of $\frac{7!}{4! \times 3!}$.

How many values of $n$ satisfy the equation $(n-1)! = n!$?

What is the remainder when $100!$ is divided by $97$?

If $n!$ has the prime factorization $2^{31} \cdot 3^{15} \cdot 5^7 \cdot 7^4 \cdot 11^2 \cdot 13^2 \dots$, find $n$.

What is the value of $\frac{0!}{1!}$?

The number of permutations of $n$ objects where $p$ objects are of one kind, $q$ of another, and the rest are distinct is given by:

Simplify $\frac{(n+1)!}{n!}$.

Which represents the number of ways to arrange $n$ objects taking all at a time?

If $^nP_r = 240$ and $^nP_{r-1} = 40$, find $n$ and $r$.

What is the value of $^nP_r$ when $r=0$ according to the derived formula?

If $^{n+r}P_2 = 110$ and $^{n-r}P_2 = 20$, find the values of $n$ and $r$.

Find the value of the positive integer $n$ that satisfies the equation $\sum_{k=0}^{r-1} \frac{^{n-k}P_{r-k}}{^{n-k}P_{r-k-1}} = 60$, where $r = 6$.

Determine the value of the positive integer $n$ if the permutation relationship $^{n+2}P_{r+1} = 132 \cdot ^{n}P_{r-1}$ holds, given the constraint $n = 2r$.

The ratio of $^{2n}P_n$ to $^{2n-1}P_n$ is simplified to:

Find the number of distinct permutations of the letters of the word $INSTITUTE$.

How many distinct arrangements can be formed from the letters of the word $DEED$?

In how many ways can the letters of the word $MAMMA$ be arranged?

Find the number of ways to arrange the letters of the word $TENNET$.

In how many ways can the letters of the word $TENNESSEE$ be arranged?

Find the number of ways to arrange the letters of the word $MUMMY$.

How many ways can $3$ identical math books, $4$ identical physics books, and $2$ identical chemistry books be arranged on a shelf?

How many arrangements of the word $BANANA$ have the three $A$'s appearing together?

How many permutations of the word $EXAMINATION$ have the vowels $A, A, I, I, E, O$ in alphabetical order?

In how many ways can the letters of the word $STATISTICS$ be arranged such that the first and last letters are the same?