In how many ways can $6$ persons be selected from $4$ officers and $8$ constables, if at least one officer is to be included

The number of ways in which we can select three numbers from $1$ to $30$ so as to exclude every selection of all even numbers is

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 :

In how many ways can $21$ English and  $19$ Hindi books be placed in a row so that no two Hindi books are together

$^n{C_r}\,{ \div ^n}{C_{r - 1}} = $

$10$ different letters of English alphabet are given. Out of these letters, words of $5$ letters are formed. How many words are formed when at least one letter is repeated