Six married couples decide to form a committee of $6$ persons. The total number of ways to form the committee such that it contains no couples is equal to:

$^{2n}C_4 : ^nC_3 = 99 : 4 \Rightarrow n = $

At an election,a voter may vote for any number of candidates,not greater than the number to be elected. There are $10$ candidates and $4$ are to be selected. If a voter votes for at least one candidate,then the number of ways in which he can vote is:

The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is:

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements is: