How many words, with or without meaning can be made from the letters of the word $MONDAY$, assuming that no letter is repeated, if.
(i) $4$ letters are used at a time.(ii) all letters are used at a time(iii) all letters are used but first letter is a vowel?

(i) There are $6$ different letters in the word MONDAY.
Number of $4$-letter words that can be formed from the letters of the word MONDAY, without repetition of letters, is the number of permutations of $6$  different objects taken $4$ at a time, which is ${}^6{P}_4$.
Thus, required number of words that can be formed using $4$ letters at a time is
${}^6{P}_4=\frac{6!}{(6-4)!}=\frac{6!}{2!}=\frac{6\times 5\times 4\times 3\times 2!}{2!}=6\times 5\times 4\times 3=360$
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= $6!=6\times 5\times 4\times 3\times 2\times 1=720$Number of vowels are $2(O,A)$
Six letters word is to be formed.
$\square \square \square \square \square \square$
First letter should be a vowel. So, the rightmost place of the words formed can be filled only in $2$ ways.
Since the letters cannot be repeated , the second place can be filled by the remaining $5$ letters. So, second place can be done in $5$ ways
Similarly, third place in $4$ ways , fourth place in $3$ ways, fifth place in $2$ ways, sixth place in 1 way.

Hence, required number of words that can be formed using four letters of the given word is $2\times 5\times 4\times 3\times 2\times 1=240$