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Let $f=x+y+xz+yw$ where $x,y,z,w$ are variables. Let $S=\{\mbox{ monomials of }f^{20}\}$. How can I calculate the product of elements of $S$ and cardinality of $S$?

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1 Answer 1

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First note that all the monomials in the expansion


are distinct. Thus there are as many of them as partitions of $20$ into four parts with zeros allowed. Of these there are $\binom{20+4-1}{4-1}=\binom{23}3=1771$.

As for the product, by symmetry each of the terms will occur the same number of times as a factor, so we only have to count how many times one term, say $x$, occurs. There are


different monomials with $n_x$ factors of $x$, so the total number of factors of each term in the product is


Thus the product is


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Thanks for the help. – user12290 Aug 21 '11 at 9:16

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