Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is

The number of $3$-$digit$ odd numbers,whose sum of digits is a multiple of $7$,is

Let $x$ denote the number of ways of selecting at least one ball from a bag containing $3$ identical red balls,$4$ identical blue balls,and $5$ identical green balls. Let $y$ denote the number of ways in which a student can fail in an examination,when they have to write the examination in $5$ different subjects. Then $x+y=$

Let $x$ and $y$ be two $2$-digit numbers such that $y$ is obtained by reversing the digits of $x$. Suppose they also satisfy $x^2-y^2=m^2$ for some positive integer $m$. The value of $x+y+m$ is

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) $TRUE$?
$(A) n_1 = 1000$
$(B) n_2 = 44$
$(C) n_3 = 220$
$(D) \frac{n_4}{12} = 420$

The value of $\sum_{r=0}^{20} {}^{50-r}C_{6}$ is equal to