The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ in which at least two digits are identical is

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

If $n$ objects are arranged in a row,then the number of ways of selecting three of these objects so that no two of them are next to each other is:

In how many different ways can the letters of the word $ALGEBRA$ be arranged in a row if:
$(I)$ The $2$ $A$s are together?
$(II)$ The $2$ $A$s are not together?

In a club election,the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote is $62$,then the number of candidates is:

There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back is: