In how many ways can a committee of $6$ members be formed out of $10$ members,such that it always includes a specified member?

The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball is:

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him 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:

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables such that at least one officer is included?

If $\binom{n}{3} + \binom{n}{4} > \binom{n+1}{3}$,then: