The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball is:

In a lottery,a person chooses six different natural numbers at random from $1$ to $20$. If these six numbers match the six numbers already fixed by the lottery committee,he wins the prize. What is the probability of winning the prize in the game? [Hint: The order of the numbers is not important.]

If a team of $4$ persons is to be selected out of $4$ married couples to play a mixed doubles tennis game,then the number of ways of forming a team in which no married couple appears 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?

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio $2:1$,then $n=$