The number of seven-digit numbers that can be formed by using the digits $1, 2, 3, 5,$ and $7$ such that each digit is used at least once is:

The difference between an integer $n$ and its cube $n^3$ is always divisible by which of the following?

The number of even numbers greater than $1000000$ that can be formed using all the digits $1, 2, 0, 2, 4, 2, 4$ is

Let $S$ denote the set of $4$-digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the set of $5$-digit numbers having the product of its digits equal to $20$. Then $n(S) + n(P)$ is equal to:

$A$ bag contains $n$ white and $n$ black balls. Pairs of balls are drawn at random without replacement successively,until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is $14400$,then $n$ is equal 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 ...... .