In an examination,there are $3$ multiple-choice questions,and each question has $4$ choices. The number of ways in which a student can fail to get all answers correct is:

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

$A$ man has $7$ relatives,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,is:

Suppose $a_2, a_3, a_4, a_5, a_6, a_7$ are integers such that $\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!}$,where $0 \leq a_j < j$ for $j = 2, 3, 4, 5, 6, 7$. The sum $a_2 + a_3 + a_4 + a_5 + a_6 + a_7$ is:

Total number of $3$ letter words that can be formed from the letters of the word $'SAHARANPUR'$ is equal to

The number of ways of selecting $15$ teams from $15$ men and $15$ women,such that each team consists of one man and one woman,is