The letters of the word "$UDAYPUR$" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "$UDAYPUR$" is:

The number of ways in which $n$ boys and $n$ girls can be arranged in a row such that all the boys are together and all the girls are also together is equal to

The number of words that can be written using all the letters of the word "$IRRATIONAL$" is

In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ must speak after both $S_1$ and $S_2$?

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

If ${}^{2n+1}P_{n-1} : {}^{2n-1}P_n = 11 : 21$,then $n^2 + n + 15$ is equal to: