Question

Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements do all vowels occur together.

Answer 1

$INDEPENDENCE$

Arranging $5$ vowels

Since vowels are coming together

they can be $EEEEI,IEEEE$, and So on

In $IEEEE$ there are $4E$

since letter are repeating total arrangement $=\frac{5!}{4!}$

Arranging Remaining

no we need to arrange

$7+1=8$

Hence are $3N,2D$

We use$=\frac{8!}{3!2!}$

Required No. $=\frac{5!}{4!}\times \frac{8!}{3!2!}$

$=\frac{(8\times 7\times 6\times 5\times 4)(5\times 4!)}{4!\times 3!\times 2}$

$=16,800$