The number of seven-digit positive integers formed using the digits $1, 2, 3,$ and $4$ only,such that the sum of the digits is equal to $12$,is $...........$.

In how many ways can $n$ prizes be distributed among $n$ boys such that no boy gets all the prizes?

Let $S_1 = \{(i, j, k) : i, j, k \in \{1, 2, \ldots, 10\}\}$,$S_2 = \{(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in \{1, 2, \ldots, 10\}\}$,$S_3 = \{(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in \{1, 2, \ldots, 10\}\}$,$S_4 = \{(i, j, k, l) : i, j, k \text{ and } l \text{ are distinct elements in } \{1, 2, \ldots, 10\}\}$. If the total number of elements in the set $S_r$ is $n_r$ for $r = 1, 2, 3, 4$,then which of the following statements is (are) $TRUE$?
$(A) n_1 = 1000$
$(B) n_2 = 44$
$(C) n_3 = 220$
$(D) \frac{n_4}{12} = 420$

For all $n \in N$,the product $(n+24)(n+25)(n+26)(n+27)$ is always divisible by:

Out of $8$ students in a classroom,$4$ of them are chosen and they are arranged around a table. If the remaining $4$ are arranged in a row,then the total number of arrangements that can be made with those $8$ students is

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $