Combinations

If ${}^{\mathrm{15}}{C}_{3r}={}^{15}{C}_{r+3}$, then the value of $r$ is

If  ${{}^{20}C}_{r+2}={{}^{20}C}_{2r-3}$, find ${{}^{12}C}_r$.

An examination paper consists of$12$ questions divided into parts$A$ &$B$. Part-$A$ contains$7$ questions & Part-$B$ contains$5$ questions. A candidate is required to attempt$8$ questions selecting$3$ at least from each part. In how many maximum ways can the candidate select the questions?

The total number of paths from the origin to the point $\left(9,9\right)$ in the cartesian plane which never pass through $\left(5,5\right)$, where paths consisting only of steps going $1$ unit North or $1$ unit East, is

The number of positive divisors of $10!$ are

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is $840$, then the total numbers of persons, who participated in the tournament, is ________.

if the coefficients of three consecutive terms in the expansion of ${\left(1+x\right)}^n$ are the ratio $1 : 5 :20$then the coefficient of the fourth term is 

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A\times B$ each having  at least $3$ and at most $6$ elements is

If ${}^{2n}C_3:{}^{n}C_3=10:1,$ then the ratio $\left(n^2+3n\right):\left(n^2-3n+4\right)$ is

If we have a ATM pin of $4$ digit.The sum of first two digit is equal to the sum of last two digits and the greatest integer used is $7$, then number of trials used to get the pin if all digits are different

If the coefficient of three consecutive terms in the expansion of ${\left(1+x\right)}^n$ are in the ratio $1:5:20$, then the coefficient of the forth term of the expansion is:

Find all the $4$ letter words with $2$ vowels and $2$ consonants from the word UNIVERSE.

If ${}^{2n}C_3:{}^{n}C_3=10$ then $\frac{n^2+3n}{n^2-3n+4}$ is equal to 

There are $12$ courses available in which $5$ courses are of languages. Find the number of ways to select $5$ courses so, that atmost $2$ languages courses are there. 

Number of ways in which $10$ different toys can be distributed among two brothers in ratio $2:3$ is

Out of $6$ books, in how many ways can a set of one or more books be chosen

$n$ and $r$ integers such that $1\le r\le n$, then $n\left({}^{n-1}C_{r-1}\right)$ is

If $C\left(2n,3\right):C\left(n,2\right)=12:1$, then $n$ is