The number of values of $'r'$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2}=^{69}C_{r^2-1} - ^{69}C_{3r}$ is :-  

A committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$ young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ young men. Then the total number of ways in which this committee can be formed is

In how many ways a team of $10$ players out of $22$ players can be made if $6$ particular players are always to be included and $4$ particular players are always excluded

There are three bags $B_1$,$B_2$ and $B_3$ containing $2$ Red and $3$ White, $5$ Red and $5$ White, $3$ Red and $2$ White balls respectively. A ball is drawn from bag $B_1$ and placed in bag $B_2$, then a ball is drawn from bag $B_2$ and placed in bag $B_3$, then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed, if same colour balls are used in first and second transfers (Assume all balls to be different) is

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

Let $n(A) = 3, \,n(B) = 3$ (where $n(S)$ denotes number of elements in set $S$), then number of subsets of $(A \times B)$ having odd number of elements, is-