All possible two-factor products are formed from the numbers $1, 2, 3, 4, \dots, 200$. The number of products out of the total obtained which are multiples of $5$ is:

In an election,a voter can vote for any number of candidates but not more than the number of candidates to be elected. There are $10$ candidates and $4$ are to be elected. If a voter must vote for at least one candidate,in how many ways can they vote?

Let $S = \{1, 2, 3, \ldots, 9\}$. For $k = 1, 2, \ldots, 5$,let $N_k$ be the number of subsets of $S$,each containing five elements out of which exactly $k$ are odd. Then $N_1 + N_2 + N_3 + N_4 + N_5 =$

The number of ways of distributing $8$ identical rings to $3$ different girls so that every girl gets at least $1$ ring is

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements 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?