Question

Find the probability in a random arrangement of the letters of the word ${}^{\prime }SOCIA{L}^{\prime }$ vowels come together.

Answer 1

We have,

There are $3$ vowels and $3$ other letters as the $3$ vowels are to be together,

Consider them as one object.

Therefore there are $4$ objects totally, which can be arranged in $4!$ways,but the vowels can be arranged internally in $3!$ ways.

Therefore total no, of ways of arranging the word with the vowels together is $4!\times 3!=24\times 6=144$

That is no, of word with vowels together is $144$.

But the total number of random arrangement are $=6!=720$

Required probability $=\frac{720}{144}=\frac{1}{5}$