I'm reviewing for a combinatorics final exam in two days... can anybody confirm or deny whether my understanding of this is correct?
Using a generating function, find the number of ways to pick 6 balls from four different piles of 3 balls.
So I've got the generating function to be $(1+x+x^2+x^3)^4$, and I'm looking for the coefficient of $x^6$... I've determined this to be
$${4\choose 1}{5\choose 2}+{9\choose 6}$$
I can't think of any way to verify if this is correct, sadly. The problem seems too complex (or I'm just forgetting how) to use the standard multipilication and addition principles.
I don't know if it's right or wrong. I'd like to find a way to verify this using another method that makes sense, or if somebody can tell me if that's right. I'm trying to think of how to do this without using generating functions (such as just using the multiplication and addition principles) but I can't think of how to do it. It's been too long
@PedroTamaroff I will have to see what it is about SNF that keeps it from being the SVD. The SVD is a diagonal times an invertible matrix on either side.
@TedShifrin Jacobson's explanation of the algorithm to get Smith's normal form is good enough I can work out some matrices, but not good enough to get a satisfactory proof.
if we define multiplication of cosets of K simply as product of group subsets, and prove its closed ( provided K is normal ), we don't need to prove that the multiplication is well-defined .. But now if we define the multiplication of cosets uK and vK as uK o vK = (uv)K we don't need to prove its closed ( since we assumed it is ), but we need to prove its well-defined ? is that how it works?
maybe i should better phrase
Given a group G, and K a subgroup of G.If we define multiplication of cosets of K simply as a product of group subsets, and prove its closed ( provided K is normal in G ), we don't need to prove that the multiplication of cosets is well-defined ..
But now if we define the multiplication of cosets uK and vK as uK o vK = (uv)K we don't need to prove its closed ( since we assumed it is ), but we need to prove its well-defined ( provided K is normal in G, or eqvalently ,the kernel of some homomorphism?
@Sawarnik, no, i do not think you will be sued for IDIOT word!!
@Sawarnik, i think modi is very good candidate, as he has really developed Gujarat, all outsiders, come here just because of booming economy of Gujarat!
Using generation functions solve the following difference equation:
$$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
How do I go about this? I don't really get difference equations
@TedShifrin Wolfram gets $ \frac{4}{3} \log ( 1 - x^{3}{4} ) + \mathcal{C} $. Is it just a different form? @Sawarnik The integral above is pretty similar. You'll need a substitution that'll remove the radicals and leave you with a rational function.
Using generation functions solve the following difference equation:
$$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
How do I go about this? I don't really get difference equations
