1. Oct 20, 2007

### tka2451

1. The problem statement, all variables and given/known data

Suppose you throw a six-sided die five times. Find the probability that the sum of the outcomes of the throws is 15 using generating functions.

2. Relevant equations

Binomial theorem, generating functions

3. The attempt at a solution

Here's my attempt:

Okay, so I know so far that the generating function of a single throw is
G(x) = s/6 + (s^2)/6 + (s^3)/6 + (s^4)/6 + (s^5)/6) + (s^6)/6.

And that G(x) raised to the 5th power is the generating function for five throws.

Also, taking into account the independence of the x's:

G_x(t) = ((s+...+s^6)/6)^5 = s^5 * (1-s^6)^5 divided by 6^5 * (1-s)^5.

I get that:

1/(1-s)^5 = 1 + (5 choose 1) s + (6 choose 2) s^2 + (7 choose 3) s^3 + (8 choose 4) s^4 + (9 choose 5) s^5

and that

(1-s^6)^5 = 1 - (5 choose 1) s^6 + (5 choose 2) s^12 - (5 choose 3) s^18 + (5 choose 4) s^24 - (5 choose 5) s^30

However, I'm confused about how I use these to figure out the coefficient of s^15, which is the probability I'm looking for.

Last edited: Oct 20, 2007