The value of ${}^6P_4 + 4 \cdot {}^6P_3$ is $.......$ .

If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition,then $\alpha: \beta$ is

The value of $(2 \cdot {}^{1}P_{0} - 3 \cdot {}^{2}P_{1} + 4 \cdot {}^{3}P_{2} - \dots \text{ up to } 51^{\text{th}} \text{ term}) + (1! - 2! + 3! - \dots \text{ up to } 51^{\text{th}} \text{ term})$ is equal to

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.

$A$ letter lock consists of three rings with $15$ different letters each. If $N$ denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then:

How many numbers can be formed from the digits $1, 2, 3, 4$ when repetition is not allowed?