When ten persons shake hands with one another,in how many ways is it possible?

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by $66$,then the number of men who participated in the tournament lies in the interval

In how many ways can the letters of the word '$LEADER$' be arranged?

How many odd numbers less than $1000$ can be formed using the digits $0, 2, 5, 7$? (repetition of digits is allowed).

How many words can be formed using the letters of the word $INDEPENDENCE$,such that all vowels always come together?

$1 + \sum\limits_{r = 0}^{22} {\left\{ {r\left( {r + 2} \right) + 1} \right\}} \cdot r! = k!$,then the number of divisors of $k$ is