Combinatorics question on the number of words that can be formed from the letters of another word, subject to some constraints.

Question

Find the number of different words that can be formed using all the letters of the word DEEPMALA if two vowels are together and the other two are also together but separated from the first two.

Attempt:

Consider EE as a unit and AA as a unit. Then we have EE, AA, P, L, M, D as 6 units. Total number of words = $6!$

Now consider EEAA as a single unit. Now we have 5 units.

Hence total number of words now = $5! \times 2 $( multiplied by 2 because AAEE is also possible)

Thus answer should be: $6! - 5!\times 2 = 480$

But answer given in my book = $1440$

What is my mistake?

Answer 1

All it said was $2$ vowels.

You simply missed the cases with $EA$ and $AE$.

Including these should give you the correct result

Answer 2

Consider EE as 1 unit and AA as 1 unit. So total number of letters=6 Number of words=6! Now consider AE as 1 unit and EA as 1 unit. So, number of words=6! Therefore, the answer is=6! x 6!=1440