$A$ student is required to answer $10$ questions out of $13$ in an examination,such that he must choose at least $4$ questions from the first $5$ questions. How many ways can he make the selection?

The English alphabet has $5$ vowels and $21$ consonants. How many words with $2$ different vowels and $2$ different consonants can be formed from the alphabet?

In how many ways can $10$ people be seated in $2$ boats such that each boat has $5$ people and two specific people do not sit in the same boat?

$A$ village has $10$ players. $A$ team of $6$ players is to be formed. $5$ members are chosen out of these $10$ players and then the captain is chosen from the remaining $5$ players. The total number of ways of choosing such a team is:

The value of $r$ for which $^{15}C_{r+3} = ^{15}C_{2r-6}$ is

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?