# How do I approach this combinatorics problem about firing order in an engine?

Here is the question:

(a) In a six-cylinder engine, the even-numbered cylinders are on the left and the odd-numbered cylinders are on the right. A good ﬁring order is a permutation of the numbers 1 to 6 in which right and left sides are alternated. How many possible good ﬁring orders are there which start with a left cylinder?

b) Repeat for a 2n-cylinder engine.

For for first part, I figured it out with enumeration. I did notice a pattern though: that you have 3 choices then 3 choices , 2 then 2, etc.

For the second part I'm a bit confused about how to do it. I appreciate any tips or advice.

-
Your pattern is a good start. What happens for an eight-cylinder engine? Try to phrase your answer in a way that generalizes the pattern you just found. Another hint is that $3 \cdot 3 \cdot 2 \cdot 2 \cdot 1 \cdot 1 = (3 \cdot 2 \cdot 1) \cdot (3 \cdot 2 \cdot 1)$. –  Michael Joyce Feb 28 '12 at 20:17
We can consider this equivalent problem : the cylinder of left stay left and right stay right. In this case you have $n!$ possibility to permute them in each side. Thus $n!^2$ for all.