The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

In how many different ways can the letters of the word '$BANKING$' be arranged so that the vowels always come together?

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee should have all $4$ professors and $1$ research associate $OR$ all $3$ trainees and $2$ professors?

If $\sum_{i=0}^4 {^{4+i}}C_i + \sum_{j=6}^9 {^{3+j}}C_j = {^x}C_y$ ($x$ is a prime number),then which one of the following is incorrect?

If $a_n = \sum_{r = 0}^n \frac{1}{^nC_r}$,then $\sum_{r = 0}^n \frac{r}{^nC_r}$ equals

If the number of five-digit numbers with distinct digits and $2$ at the $10^{\text{th}}$ place is $336k$,then $k$ is equal to