Math 344: Introduction to Combinatorics

Comments on Homework: Section 6.2

Problem 11

Question

Could you please provide some hints for part (e)?

Answer

For this, use equation (5) in table 6.1 (page 248) with n=m+1. You should find that the general term is b^(m+r) C(r+m,r) x^(m+r). When m+r=12, this coefficient simplifies to b¹² C(12,r) = b¹² C(12,m) (and the m version is more appropriate, because r is determined only by m+r=12, whereas m is given.

Problem 17

Question

I think the answer for part (c) should be C((10-2k)+2-1,(10-2k)). The book is missing the final 2.

Answer

I think you're right.

Problem 36

Question

Could you please provide some hints for part (e)?

Answer

I hope that the equality “proved” in this exercise strikes you as quite wrong.

How did we get equation (5)? It comes from expanding the series of [1/(1-x)]ⁿ = [1 + x + x^2 + ... ]ⁿ. For what x is this series valid?