The sum of the digits in the unit's place of all the $4-$ digit numbers formed by using the digits $3, 4, 5,$ and $6$,without repetition,is:

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

How many numbers less than $10000$ can be formed using the eight digits $1, 2, 3, 4, 5, 6, 7, 0$? (Repetition of digits is allowed)

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if there are always $4$ letters between $P$ and $S$?

The number of words (with or without meaning) that can be formed from all the letters of the word $LETTER$ in which vowels never come together is

If $3$ books on Physics,$2$ books on Chemistry,and $4$ books on Mathematics are to be arranged on a shelf so that all the Physics books are together and all the Mathematics books are together,then the number of such arrangements is