If the best and the worst paper never appear together,then six examination papers can be arranged in how many ways?

If $(n+2)! = 2550(n!)$,find $n$.

$A$ test consists of $6$ multiple choice questions,each having $4$ alternative answers of which only one is correct. The number of ways in which a candidate answers all six questions such that exactly four of the answers are correct is:

The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to

There are $4$ notes of Rs. $100$ and $5$ other notes of denominations Rs. $1$,Rs. $2$,Rs. $5$,Rs. $20$,and Rs. $50$. These notes are to be distributed among $3$ children such that each child receives at least one note of Rs. $100$. Find the total number of ways of distribution.

$A$ number lock consists of three rings. If the $1^{st}$ ring is marked with digits $0$ to $9$,the $2^{nd}$ ring is marked with prime numbers greater than $2$ but less than $30$,and the $3^{rd}$ ring is marked with all vowels,find the total number of unsuccessful attempts.