Team $A$ consists of $7$ boys and $n$ girls and Team $B$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl,then $n$ is equal to

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

Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example,$2210222$ is in $S$,but $0210222$ is not in $S$. Then the number of elements $x$ in $S$ such that at least one of the digits $0$ and $1$ appears exactly twice in $x$,is equal to $....$

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $x$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,

If $\frac{{}^{n + 2}C_6}{{}^{n - 2}P_2} = 11$,then $n$ satisfies the equation