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.]

There are $(n + 1)$ white and $(n + 1)$ black balls,each set numbered $1$ to $(n + 1)$. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is

If $a, b$ and $c$ are the greatest values of $^{19}C_{p}, ^{20}C_{q}$ and $^{21}C_{r}$ respectively,then

The total number of words that can be formed by using all the letters of the word $ALLEN$ such that all vowels occur in alphabetical order is:

In how many different ways can the letters of the word '$DETAIL$' be arranged in such a way that the vowels occupy only the odd positions?

$P_1$ and $P_2$ are two distinct and intersecting planes. Three non-collinear points lie on $P_1$ and another three non-collinear points lie on $P_2$ (none being on the line of intersection of the planes). Then,the maximum number of tetrahedrons formed using these six points is: