Actual responses to actual questions from students of Math 344, Introduction to Combinatorics (Spring 2004).
Help! Please give me a hint for this problem!
The question asks you to find five numbers, let's call them r1 through r5, such that
1 ≤ r1 < r2 < r3 < r4 < r5 ≤ 20.
Moreover, we also require that rk-rk-1 ≥ 2 for k=2,3,4,5. Can we rephrase this question in terms of, say, the solutions to a Diophantine equation? Here's a thought: let xk = rk-rk-1, where we set r0=0, so the problem becomes
x1 + x2 + x3 + x4 + x 5 ≤ 20,
where xk≥2 for k=2,3,4,5. How many solutions does this equation have?
The generating function given in the back of the text has six products, not five. What's up with that?
We're not used to solving inequalities like the one given above:
x1 + x2 + x3 + x4 + x 5 ≤ 20.
Instead, we've studied equalities. How can we turn the inequality we have into an equality? Just add an extra variable x6≥0, so we get
x1 + x2 + x3 + x4 + x 5 + x 6 = 20.
This doesn't give me the right answer!
Well, this gives me the answer
(x+x2+x3+···) (x2+x3+x4+···)4 (1+x+x2+x3+···)
This is not the same answer as in the book, but it is more or less equivalent. (For this equation, we're interested in the coefficient of xn, not xn-5 as the book is).