The number of permutations of $n$ things taken $r$ at a time,when $p$ specific things are always included,is

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

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

The sum of all the $4$-digit distinct numbers that can be formed with the digits $1, 2, 2,$ and $3$ is:

If all the words with or without meaning made using all the letters of the word $NAGPUR$ are arranged as in a dictionary,then the word at $315^{\text{th}}$ position in this arrangement is:

The number of ways of forming a queue of $4$ boys and $3$ girls such that all the girls are not together,is: