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 distinct terms in the expansion ${\left(x_1+x_2+x_3+.....+x_n\right)}^3$

Two numbers $a$ and $b$ are chosen at random from the set $\left\{1,2,3,,......,7n\right\}$. The probability that $a^4-b^4$ is divisible by $7$ is 

What is combination in mathematics?

If total number of students is $10$ and number of students to be selected or permutated is $5$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $6$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $15$ and number of students to be selected or permutated is $7$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $12$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

If total number of students is $8$ and number of students to be selected or permutated is $4$. Find the number of ways of selection and permutation and state the difference between permutation and combination.

Two classrooms $A$ and $B$ having capacity of $25$ and $\left(n-25\right)$ seats, respectively. Let $A_n$ denotes the number of possible sitting arrangements of room $A$, when $n$ students are to be seated in these rooms, starting from room A which is to be filled up full to its capacity. If $A_n-A_{n-1}=25!\left({}^{49}\mathrm{C}_{25}\right)$, then $n$ equals

${}^{4n}C_{2n}:{}^{2n}C_n={\left[1.3.5\dots \dots .(4n-1)]:[1.3.5\dots \dots (2n-1)\right]}^2$.

If ${}^{n}C_4=210$ then $n=$

Let $E$ denote the set of letters of the English alphabet, $V=\left\{a,e,i,o,u\right\}$ and $C$ be the complement of $V$ in $E$. Then, the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is

Find the number of ways of selecting a cricket team of $11$ players from $7$ batsman and $6$ bowlers such that there will be atleast $5$ bowlers in the team.

If ${}^{n}C_5={}^{n}C_6$, then ${}^{13}C_n=$

If ${}^{n}C_r$ denotes the number of combinations of '$n$' things taken '$r$' at a time, then the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2{}^{n}C_r$ equals