If $2 \times {}^n{C_5} = 9\,\, \times \,\,{}^{n - 2}{C_5}$, then the value of $n$ will be

The number of ways in which a committee of $6$ members can be formed from $8 $ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

If $^{15}{C_{3r}}{ = ^{15}}{C_{r + 3}}$, then the value of $r$ is

A car will hold $2$ in the front seat and $1$ in the rear seat. If among $6$ persons $2$ can drive, then the number of ways in which the car can be filled is

A total number of words which can be formed out of the letters $a,\;b,\;c,\;d,\;e,\;f$ taken $3$ together such that each word contains at least one vowel, is

An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at the most three of them are red is