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 for some $\mathrm{m}, \mathrm{n} ;{ }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{n+1} C_m$ is equal to

A person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$, then $n$ equals

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

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

A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of:

at least $3$ girls?