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We divide a group of people into subgroups A, B, and C, and ask each subgroup to form a line. We also require that A have an odd number of people, and that B have an even number of people. How many ways are there to do this?

Here is my attempt using generating functions

Odd number of people

  • $ A(x) = \sum n! (x^n/n!) = x + x^3 + x^5 + ... = \dfrac{x}{1-x^2}$

Even number of people

  • $ B(x) = \sum n! (x^n/n!) = 1 + x^2 + x^4 + ... = \dfrac{1}{1-x^2}$

No constraints on group C

$ C(x) = \sum n! (x^n/n!) = 1 + x + x^2 + ... = \dfrac{1}{1-x}$

It then follows by product rule

$G(x) = {{\dfrac{1}{1-x} * \dfrac{1}{1-x^2} * \dfrac{x}{1-x^2}}} $

Does this look correct?

thetraveller
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  • Well, the question itself isn't well defined, since there are obviously an infinite number of ways to satisfy those conditions. As such, how are you going to find the degree of $x$ in your product. This doesn't make sense. – Rushabh Mehta Oct 29 '18 at 03:14
  • @RushabhMehta I don't see how there are infinite number of ways to satisfy these conditions? we are interested in the co-efficient of $x^n$ that can be found easily by solving the above generating function using partial decomposition – thetraveller Oct 29 '18 at 03:28
  • @RushabhMehta which part didn't make sense? – thetraveller Oct 29 '18 at 03:29

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