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?

The number of words that can be formed using the letters of the word $ARTICLE$,such that the vowels occupy even places,is

$1 + \sum\limits_{r = 0}^{22} {\left\{ {r\left( {r + 2} \right) + 1} \right\}} \cdot r! = k!$,then the number of divisors of $k$ is

How many numbers lying between $99$ and $1000$ can be formed using the digits $2, 3, 7, 0, 8, 6$,if each digit can be used only once in each number?

The number of triangles that can be formed by choosing the vertices from a set of $12$ points,seven of which lie on the same straight line,is

$A$ 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.