$^n{C_r}{ + ^{n - 1}}{C_r} + ......{ + ^r}{C_r}$ =

${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if:

The number of arrangements that can be formed from the letters $a, b, c, d, e,f$ taken $3$ at a time without repetition and each  arrangement containing at least one vowel, is

If $^{2n}{C_3}:{\,^n}{C_2} = 44:3$, then for which of the following values of $r$, the value of $^n{C_r}$ will be 15

How many words can be formed by taking $3$ consonants and $2$ vowels out of $5$ consonants and $4$ vowels

Total number of $6-$digit numbers in which only and all the five digits $1,3,5,7$ and $9$ appear, is