Question

How many words, with or without meaning can be formed 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 vowel?

Or

How many different words can be formed by using all the letters of the word 'ALLAHABAD' ?

(i) In how many of them, vowels occupy the even position ?

(ii) In how many of them, both 'L' do not come together ?

Answer 1

There are 6 letters in word 'MONDAY'

(i) The total number of 4 letter words formed from the letter of the word "MONDAY" = ${}^6{C}_4\times 4!$

$=\frac{6!}{2!4!}\times 4!$

$\frac{6!}{2!}=360$

(ii) The total number of words formed using all letters of the word 'MONDAY' = 6! = 720

(iii) There are two vowels A and O. So, first place can be filled in 2 ways and the remaining 5 places can be filled in 5! ways.

So, total number of words beginning with a vowel = $2\times 5!=2\times 120=240$

Or

There are 9 letters in word 'ALLAHABAD' out of which 4 are A's, 2 are L's and the rest are all distinct.

So, the total number of words formed

= $\frac{9!}{4!2!}=7560$

(i) There are 4 vowels and all are alike, i.e. 4A's. Also, there are 4 even places can be occupied by 4 vowels in $\frac{4!}{4!}$ = 1 way. Now, we are left with 5 places in which 5 letters of which two are alike (2L's) and other distinct can be arranged in $\frac{5!}{2!}=60$ ways.

Hence, total number of words in which vowels occupy the even places = $\frac{5!}{2!}\times \frac{4!}{4!}$ = 60 ways.

(ii) Considering both L together and treating them as one letter, so we have 8 letters out of which A repeats 4 times and other are distinct. These 8 letters can be arranged in $\frac{8!}{4!}=1680$

So, the total number of words in which both L come together

= $\frac{8!}{4!}=1680$

Hence, the number of words in which both L do not come together

= Total number of words - Number of words in which both L come together

= 7560 - 1680 = 5880