If $(2 \leq r \leq n)$, then ${}^{n}C_{r} + 2 \cdot {}^{n}C_{r+1} + {}^{n}C_{r+2}$ is equal to:

If at the end of a certain meeting,everyone had shaken hands with everyone else,and it was found that $45$ handshakes were exchanged,then the number of members present at the meeting is:

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?

$A$ student has to answer $10$ questions,choosing at least $4$ from each of the parts $A$ and $B$. If there are $6$ questions in part $A$ and $7$ in part $B$,then the number of ways the student can choose $10$ questions is:

The number of ways of choosing a committee from $4$ men and $6$ women so that the committee includes at least $2$ men and exactly twice as many women as men is

Everybody in a room shakes hands with everybody else. The total number of handshakes is $66$. The total number of persons in the room is