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?

Explanation

There are 6 different letters in the word MONDAY.

(i) 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`

(ii) Number of words that can be formed by using all the letters of the word MONDAY at a time is the number of permutations 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`

(iii) In the given word, there are 2 different vowels, which have to occupy the rightmost place of the words formed. This can be done only in 2 ways.

Since the letters cannot be repeated and the rightmost place is already occupied with a letter (which is a vowel), the remaining five places are to be filled by the remaining 5 letters. This can be done in` 5 !` ways.

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