Question

In how many ways can 10 different prizes be given to five students if the head boy must get 4 prizes and rest of the students can get any number of prizes?

Answer 1

Out of 10 prizes we have to select four for the head boy which can be done in ${}^{10}{C}_4$ ways. Now we are left with 6 prizes and 4 other boys. Each prize can be given in 4 ways. Hence the
remaining 6 prizes can be given in 46 ways.
By fundamental theorem total ways
$={}^{10}{C}_4\times 4^6=210\times 4096=8601060.$