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Question

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?

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Solution

(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 6P4.
Thus, required number of words that can be formed using 4 letters at a time is
6P4=6!(64)!=6!2!=6×5×4×3×2!2!=6×5×4×3=360
(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 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 6×5×4×3×2×1=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 6P6=6!
Thus, required number of words that can be formed when all letters are used at a time= 6!=6×5×4×3×2×1=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 2×5×4×3×2×1=240

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