No. of different possible arrangements.

Question

How can I find no. of different possible arrangements with the factor of the term $a^2b^4c^5$ written at full length.

Answer 1

There are $2$ $a's$, $4 \ b's$ and $5 \ c's$.And you want to count number of strings of length $2+4+5 = 11$. Select the positions for a's in $11 \choose 2$ ways. Select the positions for b's from remaining $9$ positions in $9 \choose 4$ ways. You must place c's in the remaining positions. So, Total = ${11 \choose 2}*{9 \choose 4} $

Answer 2

It's just $$\binom{11}{2, 4, 5}={11!\over2!4!5!}.$$

Answer 3

Use the multinomial distribution. You have $11$ terms. Permute them in $11!$ ways. Then divide out by the number of ways you can permute each individual character in a given term. So our answer is:

$$ \frac{11!}{2!4!5!}$$

Notice in a given arrangement, we can re-arrange our $a$'s in two ways. We can also rearrange our $b$'s in $4!$ ways to keep the same term.

Does this make sense?