Find the number of different signals that can be generated by arranging at least $2$ flags in order (one below the other) on a vertical staff,if five different flags are available.

If $a_n = \sum_{r=0}^n \frac{1}{{}^n C_r}$,then $\sum_{r=0}^n \frac{r}{{}^n C_r} = $

The sum of the four-digit even numbers that can be formed with the digits $0, 3, 5, 4$ without repetition is:

The number of $5$ digit telephone numbers having at least one of their digits repeated is

If $P(n, r) = 1680$ and $C(n, r) = 70$,then $69n + r! = \dots$.

If four letters are chosen from the letters of the word $ASSIGNMENT$ and are arranged in all possible ways to form $4$-letter words (with or without meaning),then the total number of such words that can be formed is: