Math 344: Introduction to Combinatorics

Comments on Homework: Section 5.3

Problem 12

Question

Could you please explain what the rth order partial derivative means? I am assuming that rth order means the degree of f.

Answer

No, rth order means the number of derivatives. For example, the second partial derivatives of f(x,y) are (using "d" for "partial"):

d² f / dx², d² f / dxdy, d²f / dy²

(Note that d²f / dydx = d²f / dxdy. This is a big hint for the problem.)

Question

So if there is x_n's or n objects with repetition with r types then the answer to this question is (by Theorem 2) C(r+n-1,r)

Is this correct or do i have to get a numerical value for this?

Answer

Your answer is correct (you can't very well get a number out of the letters r and n), but you should make sure you understand how to explain it.

Problem 25

Question

Also for q 25) I am unclear as to what the notion implies when it is summing theorem 1 such that it equals 3¹⁰. Could you clarify this more?

Answer

I'm not sure I understand your question. The sum in the question is over all possible triples k₁, k₂, k₃ such that k₁ + k₂ + k₃ = 10. You could use theorem 1, but the algebra is awful. How about thinking combinatorially: what does the left-hand side mean? What does the right-hand side mean?