In how many different ways can the letters of the word '$PRAISE$' be arranged?

In how many ways can $10$ balls be divided between two boys,if one receives two balls and the other receives eight balls?

In how many ways can a committee consisting of $5$ men and $6$ women be formed from $8$ men and $10$ women?

$A$ 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 $7$-digit numbers which can be formed using the digits $1, 2, 3, 2, 3, 3, 4$ is:

There are $15$ persons in a party and each person shakes hands with every other person exactly once. What is the total number of handshakes?