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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.

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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
How would you do this problem using generating functions? I'm using the ordinary generating function but can not get an answer of 36. – tamefoxes Jun 21 '14 at 16:31
up vote 2 down vote accepted

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.

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I believe the answer is $2\cdot n!^2$, as one must first decide whether to start with an even or an odd number. – Holdsworth88 Jul 25 '15 at 18:41

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