Evaluate $\frac{n!}{(n-r)!}$ when $n=6$ and $r=2$.

How many $2$ digit even numbers can be formed from the digits $1, 2, 3, 4, 5$ if the digits can be repeated?

$A$ 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.

The number of seven-digit odd numbers that can be formed using all the seven digits $1, 2, 2, 2, 3, 3, 5$ is $.......$

Find the number of different $8$-letter arrangements that can be made from the letters of the word $DAUGHTER$ such that all vowels do not occur together.

How many words can be formed using any $3$ letters from the word $UNIVERSAL$?