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September 7

What does this expression count?

Hi. I'm working on a combinatorial problem, and I've encountered a lot of expressions with form similar to the following: ${\tbinom {9}{4}}-{\tbinom {8}{3}}+{\tbinom {7}{2}}-{\tbinom {6}{1}}+{\tbinom {5}{0}}$ . Can anyone think of a situation where this counts something? If I can think about a real-world representation of it, I might be able to count that same quantity with a different technique, and thus establish some kind of identity.

Thanks in advance for any ideas. -GTBacchus^(talk) 02:06, 7 September 2010 (UTC)[reply]

You can often simplify that kind of expression mechanically with WZ theory, if that's of any interest. See the A=B book (online) linked in the references to that article. For your concrete example I'd put a few terms into OEIS and see what comes out. 67.122.211.178 (talk) 06:11, 7 September 2010 (UTC)[reply]

For example, you have this. -- Meni Rosenfeld (talk) 09:18, 7 September 2010 (UTC)[reply]

It looks like some form of the Inclusion-exclusion principle to me. 198.161.238.19 (talk) 18:02, 7 September 2010 (UTC)[reply]

What about the number of subsets S of {1,...,9} with card(S)=5 and such that max(S) has the same parity of 9?--pm a 20:05, 7 September 2010 (UTC)[reply]

Good grief, not another homework question

If f is continuous on [0,2], and f(0) = f(2), prove that there is a real number x ∈ [1,2] such that f(x) = f(x-1).

Intuitively I know this should be true, but that hasn't helped me out much. The only approach I can think of is to show that you can find infinitely many pairs of numbers such that f(a) = f(b), and that a-b will range from 2 to 0, but this hasn't helped. Can anyone help me? 74.15.136.172 (talk) 23:02, 7 September 2010 (UTC)[reply]

Consider the function g on [1,2] with g(x)=f(x)-f(x-1). Algebraist 23:10, 7 September 2010 (UTC)[reply]

Got it, thanks! 74.15.136.172 (talk) 00:17, 8 September 2010 (UTC)[reply]

Dixon's Method of Factorization

Perhaps I'm being slow here, but under 'Method' on Dixon's factorization method, why is it the case that N = gcd(a − b, N) × gcd(a + b, N)? Surely N -divides- gcd(a − b, N) × gcd(a + b, N), since gcd(A,BC) divides gcd(A,B)gcd(A,C), but why must the equality hold here? Is it something to do with the method of computing a and b? Or perhaps I'm missing something?

Cheers, 92.40.243.116 (talk) 23:24, 7 September 2010 (UTC)[reply]

The equality is wrong, unless a + b and a − b are coprime (take N = lcm(a + b,a − b) for a counterexample).—Emil J. 10:03, 8 September 2010 (UTC)[reply]