Each of the $10$ letters $A,H,I,M,O,T,U,V,W$ and $X$ appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three letters computer passwords can be formed (no repetition allowed) with at least one symmetric letter ?

It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

In how many ways can a committee be formed of $5$ members from $6$ men and $4$ women if the committee has at least one woman

$^n{C_r}\,{ \div ^n}{C_{r - 1}} = $

All the five digits numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

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