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

In how many ways a team of $11$ players can be formed out of $25$ players, if $6$ out of them are always to be included and $5$ are always to be excluded

A  man $X$  has $7$  friends, $4$  of them are ladies and  $3$ are men. His wife $Y$ also has $7$ friends, $3$ of  them are  ladies and $4$ are men. Assume $X$ and $Y$ have no comman friends. Then the total number of ways in which $X$ and $Y$ together  can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is :

If $2 \times {}^n{C_5} = 9\,\, \times \,\,{}^{n - 2}{C_5}$, then the value of $n$ will be

A student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first five questions. The number of choices available to him is

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw