If $^{20}C_{n+2} = ^nC_{16}$,then the value of $n$ is

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

If $\binom{a^2+a}{3} = \binom{a^2+a}{9}$,then $a = \dots$

There are $15$ people at a party and each person shakes hands with every other person. What is the total number of handshakes?

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

Let $A_1, A_2, \dots, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, \dots, 11$. One hundred gold coins were won by the team in the final match. These coins are to be distributed among the players such that each player $A_i$ gets at least $i+1$ coins,except for the captain and vice-captain who get at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. If the captain is $A_1$ and the vice-captain is $A_2$,in how many different ways can these coins be distributed?