1. ## Counting

10. In a 6-cylinder automobile engine, the even numbered cylinders are on the engine’s left side and the odd numbered cylinders are on the right side. (Drawing a picture will help you to visualize this.) A good firing order is a permutation of the numbers 1 to 6 in which the right and left sides are alternated.
(a) How many possible good firing orders are there? (b) Repeat this problem for an engine with 2n cylinders.

Any thoughts? Thanks.

2. Hi cgiulz.

Originally Posted by cgiulz
10. In a 6-cylinder automobile engine, the even numbered cylinders are on the engine’s left side and the odd numbered cylinders are on the right side. (Drawing a picture will help you to visualize this.) A good firing order is a permutation of the numbers 1 to 6 in which the right and left sides are alternated.
(a) How many possible good firing orders are there?
There are 3 cylinders on the left side and 3 cylinders on the right side.

firing a cylinder on the right side, there are 3 possibilities. After that you have to fire one on the left side. there are 3 possibilities, too.
Now you are back on the right side, there are 2 possibilities (because one cylinder already is fired) - on the left side there are 2 possibilities.

Therefore

Probability("firing right, firing left, firing right, left, right left") = 3*3*2*2*1*1

I guess you can start firing a cylinder on the left side, too, therefore

Probability("firing left, right, left right left, right= 3*3*2*2*1*1

and the solution therefore is

2*(3*3*2*2*1*1)

Originally Posted by cgiulz
(b) Repeat this problem for an engine with 2n cylinders.

Any thoughts? Thanks.
Do you have any thoughts on (b) now?

Rapha

3. 2*(n^2*(n-1)^2....1*1).

Thanks, you've been super helpful .