0
$\begingroup$

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.

  • 0
    How would you do this problem using generating functions? I'm using the ordinary generating function but can not get an answer of 36.2014-06-21

1 Answers 1

2

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.

  • 0
    I believe the answer is $2\cdot n!^2$, as one must first decide whether to start with an even or an odd number.2015-07-25