If $P(n, r) = 1680$ and $C(n, r) = 70$,then $69n + r! = \dots$.

Let $n_1 < n_2 < n_3 < n_4 < n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$. Then the number of such distinct arrangements $(n_1, n_2, n_3, n_4, n_5)$ is

In how many ways can $5$ distinct balls be distributed among $3$ persons such that each person receives at least one ball?

There are $5$ students in class $10$,$6$ students in class $11$,and $8$ students in class $12$. If the number of ways,in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100k$,then $k$ is equal to $......$

Number of all possible words (with or without meaning) that can be formed using all the letters of the word $CABINET$ in which neither the word $CAB$ nor the word $NET$ appear is

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is