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.

Find the number of diagonals of a polygon with $12$ sides.

Find the number of ways of giving away $4$ similar coins to $5$ boys if each boy can be given any number(less than or equal to $4$) of coins.

Prove that $\frac{{}^{4n}C_{2n}}{{}^{2n}C_n}=\frac{1.3.5\dots (4n-1)}{\{1.3.5\dots ..(2n-1){\}}^2}$

If a set has $12$ elements, find the number of subsets of $A$ having $4$ elements.

If a set has $12$ elements, find the number of subsets of $A$ having at least $3$ elements.

If a set has $12$ elements, find the number of subsets of $A$ having at most  $3$ elements.

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