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Homework Statement
What is the coeff of [tex]x^{99}[/tex] in (x-1)(x-2)...(x-100)
2. The attempt at a solution
This has to do with the binomial coeff. I dont know how to go about it.
There is an easier way.Look at how a product develops as you add more terms,
if
[tex](x-1)(x-2)...(x-n) = x^n - (1+2+...+n)x^{n-1} + ... + (-1)^{n}*1*2*...*n [/tex]
then
[tex](x-1)(x-2)...(x-n)(x-(n+1)) = x^{n+1} - (1+2+....+n+n+1)x^{n} + ... + (-1)^{n+1}*1*2*...*n*(n+1) [/tex]
If we let n + 1 = m, then
[tex](x-1)(x-2)...(x-m) = x^m - (1+2+...+m)x^{m-1} + ... + (-1)^{m}*1*2*...*m [/tex]