- #1
Mr Davis 97
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- 44
Homework Statement
Show that ##\displaystyle \sum_{i=0}^{100} {100\choose i}{200-i\choose 198-i}x^i## is divisible ##(x+1)^{98}##.
Homework Equations
The Attempt at a Solution
I am pretty stumped, but I have a few general. I think that the the binomial theorem will be involved. That is, I think we will have to use the fact that ##\displaystyle (x+1)^{98} = \sum_{i=0}^{98} {98\choose i}x^i##. Also, I think that we might have to use a combinatorial identity to simplify the first expression, but I am not sure... Another idea of mine was to try to show that the remainder must be zero, but then I see that this would involved a polynomial of degree 97 or less, which doesn't help very much with simplifying the problem... I think I just need to be nudged in the right direction.