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

Prove that for $3⩽r⩽n$.

${}^{(n-3)}C_r+3.{}^{(n-3)}C_{(r-1)}+3{.}^{(n-3)}{C}_{(r-2)}$$+{}^{(n-3)}C_{(r-3)}={}^{n}C_r$.

Find the value of ${}^{10}C_5+2·{}^{10}C_4+{}^{10}C_3$.

Simplify ${}^{34}C_5+\sum _{r=0}^4{}^{(38-r)}C_4$.

In a class there are $30$ students. If each student plays a chess game with each of the other student then find the total number of chess games played by them.

Find the number of ways of selecting $3$ girls and $3$ boys out of $7$ girls $6$boys.

Find the number of ways of selecting a committee of $6$ members out of $10$ members always including a specified member.

Find the number of ways of selecting $5$ books from $9$ different mathematics books such that a particular book is not included.

Find the number of ways of selecting $3$ vowels and $2$ consonants from the letters of the word EQUATION.