Question

The number of words, with or without meaning, that can be formed by taking $4$ letters at a time from the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is

Answer 1

Find the number of words that satisfy the given arrangement conditions

The word SYLLABUS has the following letters,

SS,Y,LL,A,B,U. (Five different letters, two repeated twice)

We need $4$ letter words with $2$ similar and $2$ dissimilar letters.

The two similar letters can be chosen in ${}^{2}C_1$ ways.

The two dissimilar letters can be chosen in ${}^{5}C_2$ ways.

Now the four letters can be arranged in $\frac{4!}{2!}$ ways.

Hence, the total number of words satisfying the given arrangement $={}^{2}C_1\times {}^{5}C_2\times \frac{4!}{2!}$

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

$=240$

Hence, $240$ four letter words are formed such that two of the letters are alike and two are different.