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

The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

How many numbers consisting of $5$ digits can be formed in which the digits $3, 4$ and $7$ are used only once and the digit $5$ is used twice?

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that each team includes exactly $3$ bowlers and $1$ wicket-keeper?

In how many different ways can the letters of the word '$PRAISE$' be arranged?