# How many different ways of planting the flowers? Peter's neighbour, Paul likes tulips. He would lik

How many different ways of planting the flowers?
Peter's neighbour, Paul likes tulips. He would like to plant 2 white, 5 red and 6 black tulips in a row in a way such that a red and a black tulip cannot be next to each other. How many different ways can he design the row?
I count two cases:
1. …w…w…
bwrwb in which wrw takes 7 positions and it can start from 7 different places.
rwbwr wbw takes 8 positions and can start from 6 different places.
2. …ww…
red ww black
black ww red Which is 2 ways
So in Total I got $7+6+2=15$ ways. But the answer is 33 ways. I can’t think of any other ways of arrangement…
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Step 1
As there are only two white tulips that can act as separators between black and red tulips, there are two possibilities
1) First place all black tulips together and all red tulips together. There are only two ways to do so -
BBBBBBRRRRR OR RRRRRBBBBBB
Now we must place one white tulip between them as separator. The other white tulip can be in any of the 12 places between them or at either end.
That leads to $2\cdot 12=24$ ways.
Step 2
2) We have all black tulips together but reds are on both sides or vice versa. In this case we need two separators and both white tulip positions get fixed.
When we have BBBBBB together, any number of red tulips between $1-4$ can be on the left and the remaining to the right. When we have RRRRR together, any number of black tulips between $1-5$ can be on the left and the remaining to the right.
That adds to $4+5=9$ ways.
Total number of ways $=24+9=33$.