There are 4 identical oak trees, 2 identical maple trees and 3 identical pine trees. How many arrangements start with maples?

Question

I've been self studying probability and I'm confused. How do you put maples in the first position and then calculate the arrangements?

Answer 1

You have in total $9$ trees, but if the first one is a maple tree then you still have $8$ trees to place in any way you want, and that will be $8!$ ways.

Also because the oak and pine trees are identical so you divide by $4!×3!$ (because it doesn't differ if the oak number $1$ is before oak number $2$ or oak number $2$ is before oak number $1$ since they are identical)

The total answer of ways will be $\frac{8!}{3!×4!}=280$

Answer 2

We can only choose the order of second to ninth trees. So we need to sort 4 oak trees, 1 maple tree and 3 pine trees. There are $\frac{8!}{3!1!4!}=280$ ways of ordering the trees.

Answer 3

Place a maple tree as first.

Then $4$ identical oak trees, $1$ maple tree and $3$ identical pine tree must follow.

Taking into account that the trees of one sort are identical there are $$\frac{8!}{4!1!3!}$$possibilities for that.