How many different signals can be transmitted by arranging $3$ red,$2$ yellow and $2$ green flags on a pole? [Assume that all the $7$ flags are used to transmit a signal.]

$A$ committee of $12$ is to be formed from $9$ women and $8$ men in which at least $5$ women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively

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:

The number of ways in which any four letters can be selected from the word $CORGOO$ is

For $2 \le r \le n$,the expression $\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to:

In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?