The number of words that can be formed using the letters of the word $ARTICLE$,such that the vowels occupy even places,is

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

There are $10$ different letters of the English alphabet. If words of $5$ letters are formed using these letters,how many words can be formed such that at least one letter is repeated?

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

If $a, b$ and $c$ are the greatest values of $^{19}C_{p}, ^{20}C_{q}$ and $^{21}C_{r}$ respectively,then

$\binom{n}{n-r} + \binom{n}{r+1}$,whenever $0 \le r \le n-1$,is equal to