Question

Find the number of ways permuting the letters of the word $PICTURE$ so that all the vowels come together.

Answer 1

Consider the word $PICTURE$, the vowels are $EIU$. Consider the $EIU$ to be a single object. Hence total number of remaining letters $=7-3=4$. The total number of object to be permuted, $4+1$ (vowel group)$=5$. Hence number of ways $5$ object can be re-arranged is
$=5!$. However, the $EIU$ will internally permute in $3!$ ways.
Hence the required number of ways
$=5!\times 3!$
$=120\times 6$
$=720$