In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ specific courses are compulsory for every student?

In an examination of $9$ papers,a candidate is successful only if they pass in more papers than they fail. In how many ways can the candidate be unsuccessful?

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$

If $2 \times {}^nC_5 = 9 \times {}^{n-2}C_5$,then the value of $n$ is:

There are $15$ players in a cricket team,out of which $6$ are bowlers,$7$ are batsmen,and $2$ are wicketkeepers. The number of ways a team of $11$ players can be selected from them so as to include at least $4$ bowlers,$5$ batsmen,and $1$ wicketkeeper is $.....$

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is