There are $15$ persons in a party and each person shake hand with another, then total number of hand shakes is

The value of $r$ for which $^{20}{C_r}^{20}{C_0}{ + ^{20}}{C_{r - 1}}^{20}{C_1}{ + ^{20}}{C_{r - 2}}^{20}{C_2} + ...{ + ^{20}}{C_0}^{20}{C_r}$ is maximum 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 $6$ element is :

The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is

Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$

The number of triplets $(x, y, z)$. where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is