$P, Q, R$ and $S$ have to give lectures to an audience. The organiser can arrange the order of their presentation in ............. ways.

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

In how many different ways can the letters of the word '$CORPORATION$' be arranged so that the vowels always come together?

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

How many words of $4$ letters beginning with '$A$' or '$E$' can be formed with the letters of the word '$EQUATOR$'?

The principal wants to arrange $5$ students on the platform such that the boy $SUNIL$ occupies the second position and such that the girl $GITA$ is always adjacent to the girl $NITA$. How many such arrangements are possible?