Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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 at 16:31
add comment

1 Answer 1

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.