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 xk in the polynomial
f(x) = (x1 + x2 + ··· + x25)4
There is a common factor of x4 in this expression so
f(x) = (x1 + x2 + ··· + x25)4
= x4 (1 + x1 + ··· + x24)4
= x4 [ (1 - x25) /(1 - x) ]4
= x4 ( 1 - x25 )4 ( 1/1 - x )4
=
x4 ( 1 - x25 )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 nCr. The number of ways of selecting r things from n things is
where , for any positive integer k, k! = k (k-1) (k-2) ··· 2 1
So, for example
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Now, back to expanding using the binomial theorem.
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This is an infinite sum but, inside the square brackets we want the coefficient of xk-4 where 0 ≤ k-4 ≤ 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 x50 is
Andrei and Shawn
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