How many words with or without meaning can be formed using the letters of the word MONDAY assuming that no letters 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.

Hint: This is a question of permutation and combination, the formulas which may be of use are \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\] These questions are mainly logic based so it's based to decode the question before doing any attempt.

Complete step by step answer:
(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
\[\begin{array}{l}
\therefore {}^6{P_4}\\
 = \dfrac{{6!}}{{(6 - 4)!}}\\
 = \dfrac{{6!}}{{2!}}\\
 = \dfrac{{6 \times 5 \times 4 \times 3 \times 2!}}{{2!}}\\
 = 6 \times 5 \times 4 \times 3\\
 = 360
\end{array}\]
(ii) 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 places 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 \[{6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720}\]
(iii) Total number of letters in the word MONDAY is 6.
Number of vowels are 2(O,A)
Six letters word is to be formed.
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 \[2 \times 5 \times 4 \times 3 \times 2 \times 1 = 240\]

Note:
The second question could also be done just by using a permutation formula as 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 \[{}^6{P_6} = 6!\].