If $n$ is an integer with $0 \leq n \leq 11$,then the minimum value of $n!(11-n)!$ is attained when a value of $n$ equals to

$A$ number is called a palindrome if it reads the same backward as well as forward. For example,$285582$ is a six-digit palindrome. The number of six-digit palindromes,which are divisible by $55$,is ...... .

If $4$-digit numbers greater than $5,000$ are randomly formed from the digits $0, 1, 3, 5,$ and $7$,what is the probability of forming a number divisible by $5$ when the repetition of digits is not allowed?

Two packs of $52$ cards are shuffled together. The number of ways in which a man can be dealt $26$ cards so that he does not get two cards of the same suit and same denomination is

In how many ways can the letters of the word $ASSASSIN$ be arranged so that no two $S$s occur together?

There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?