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.

Answer 1

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 \(^6P_4\).
Thus, required number of words that can be formed using 4 letters at a time is 
\(^6P_4=\frac{6!}{\left(6-4\right)!}=\frac{6!}{2!}\)

\(=\frac{6\times5\times4\times3\times2!}{2!}\)
\(=6\times5\times4\times3=360\)

(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 \(^6P_6=6!\)
Thus, required number of words that can be formed when all letters are used at a time = \(6! = 6\times5 \times 4 \times 3 \times 2 \times1 = 720\)

(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.
Thus, in this case, required number of words that can be formed is 
\(5! \times 2 = 120 \times 2 = 240\)