Here's the direct approach - it's just some common series index manipulations:

I'll write for "n choose a".

, so

(now substitute r = i+1 in the 2nd sum)

(now peel off the r=10)

(note: C(9, 9) = 1)

(now peel off the i=0, set r = i)

(note signs in 2nd sum)

(note: C(9,0) = 1)

(now combine the sums)

.

Thus .

Now simplify

.

Thus .

Note that evaluating at .

Thus (treat the leading 1 as r=0).

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Note that your book was wrong to exclude the , since is a 10th degree polynomial, but the answer your book gave was only a 9th degree polynomial.

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