$6$ different letters of an alphabet are given. Words with four letters are formed from these given letters. Then the number of words which have atleast one letter repeated and no two same letters are together, is

Out of $10$ white, $9$ black and $7$ red balls, the number of ways in which selection of one or more balls can be made, is

The number of ways in which any four letters can be selected from the word ‘$CORGOO$’ is

If the number of five digit numbers with distinct digits and $2$ at the $10^{\text {th }}$ place is $336 \mathrm{k}$, then $\mathrm{k}$ is equal to

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 :

The value of ${}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}{C_3}} $ is