Hi Dan,
On a roll each die will show a number between 1 and 25. Let k be the sum of the four numbers that appear on the dice, then 4 ≤ k ≤ 100.
We solved this using generating functions, so for each k we need the we need the coefficient of x^{k} in the polynomial
f(x) = (x^{1} + x^{2} + ··· + x^{25})^{4}
There is a common factor of x^{4} in this expression so
f(x) = (x^{1} + x^{2} + ··· + x^{25})^{4}
= x^{4} (1 + x^{1} + ··· + x^{24})^{4}
= x^{4} [ ^{(1  x25)} /_{(1  x)} ]^{4}
= x^{4} ( 1  x^{25} )^{4} ( ^{1}/_{1  x} )^{4}
=
x^{4} ( 1  x^{25} )^{4} (1  x)^{4}
Next we expanded using the binomial theorem but first we should say something about notation. The symbol is called a binomial coefficient and it is the number of ways of selecting r things from n things. Some textbooks use the notation _{n}C_{r}. The number of ways of selecting r things from n things is
where , for any positive integer k, k! = k (k1) (k2) ··· 2 1
So, for example
Now, back to expanding using the binomial theorem.
This is an infinite sum but, inside the square brackets we want the coefficient of x^{k4} where 0 ≤ k4 ≤ 96.
We get the coefficient to be
Here we have used the convention that
if r < 0 then = 0 and if r = 0 then = 1.
Thus, for example, if k = 50 then the coefficient of x^{50} is
Andrei and Shawn
