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

Total number of letters in word MONDAY = 6

Number of vowels in word MONDAY = 2

(i) Number of letters used = 4

$\therefore$ Number of permutations = ${}^6{P}_4$

=$\frac{6!}{(6-4)!}=\frac{6!}{2!}$

=$\frac{6\times 5\times 4\times 3\times 2!}{2!}=360$

(ii) Number of letters used = 6

$\therefore$ Number of permutations = ${}^6{P}_6$

= $\frac{6!}{0!}=6\times 5\times 4\times 3\times 2\times 1=720$

(iii) Here the first letter is vowel.

$\therefore$ Number of permutation of vowel = ${}^2{P}_1$

= $\frac{2!}{1!}=2$

Now the remaining five places can be filled with remaining five letters.

$\therefore$ Number of permutations = ${}^5{P}_5$

= $\frac{5!}{0!}=5\times 4\times 3\times 2\times 1=120$

Thus total number of permutations= $2\times 120=240.$