If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

$A$ letter lock consists of three rings with $15$ different letters each. If $N$ denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then:

The number of different words that can be formed from the letters of the word $SUCCESS$ in which the two $C$ are together but no two $S$ are together are:

The number of four-letter words that can be formed using the letters of the word $BARRACK$ is

The number of integers lying between $1000$ and $10000$ such that the sum of all the digits in each of those numbers becomes $30$ is: