If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

In how many different ways can the letters of the word '$THERAPY$' be arranged so that the vowels never come together?

The Chief Ministers of $11$ States of India meet to discuss the language problem. In how many ways can they seat themselves at a round table so that the Punjab and Madras Chief Ministers sit together?

$\sum_{0 \le i < j \le n} i \binom{n}{j}$ is equal to

If $12 \cdot {}^{n}C_{2} = {}^{2n}C_{3}$,find $n$.

In how many ways can $6$ apples be distributed among $3$ boys,there being no restriction to the number of apples each boy may get?