Question

All letters are used at a time.

Answer 1

There are $6$ different letters in the word MONDAY.
The first place can be filled in $6$ ways.
Second place can be filled by any one of the remaining $5$ letters. So, second place can be filled in $5$ ways
Third place can be filled by any one of the remaining $4$ letters. So, third place can be filled in $4$ ways
So, on continuing, number of ways of filling fourth place in $3$ ways , fifth place in $2$ ways, six place in 1 way.
Therefore, the number of words that can be formed using all the letters of the word MONDAY, using each letter exactly once is $6\times 5\times 4\times 3\times 2\times 1=6!$
Alternative Method:
Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutation of $6$ different objects taken $6$ at a time, which is ${}^6{P}_6=6!$
Thus, required number of words that can be formed when all letters are used at a time= $6!=6\times 5\times 4\times 3\times 2\times 1=720$