Everybody in a room shakes hand with everybody else. The total number of hand shakes is $66$. The total number of persons in the room is
$11$
$12$
$13$
$14$
If $^n{P_r} = 840,{\,^n}{C_r} = 35,$ then $n$ is equal to
How many $6 -$ digit numbers can be formed from the digits, $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated?
A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no comman friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is :
Out of $6$ books, in how many ways can a set of one or more books be chosen
$^n{P_r}{ \div ^n}{C_r}$ =