different ways to arrange the word ARRANGEMENT in 5 words.

Question

The total different ways that can the letters in the word

"ARRANGEMENT"

to be arranged in 11 letters is:

$$\frac{11!}{2!\times2!\times2!\times2!}=2494800\space ways $$

But how many different ways can the letters be arranged in only 5 letters?

Answer 1

The $$\frac{11!}{2!^4}$$ part is already right.

To divide it in 5 words, you have to multiply by:

$${{6 + 4}\choose{4}}$$

Given that words have at least 1 letter, you have $11 - 5 = 6$ letters left to divide over 5 buckets for 0 or more letters.

This is typically done with an eggs in baskets calculation with bars: you have to place 4 bars between the eggs (letters in this case). Out of 10 positions, you have to select 4 bars.