If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

In an election,there are $5$ candidates and $3$ vacancies. $A$ voter can vote for a maximum of $3$ candidates. In how many ways can a voter cast their vote?

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

How many different nine-digit numbers can be formed from the digits of the number $223355888$ by rearranging the digits so that the odd digits occupy even places?

In a group of $6$ boys and $4$ girls,four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

$A$ 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 common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are in this party,is: