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Antti H S Laaksonen 2017-04-20 23:20:52 +03:00
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chapter22.tex
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@ -112,8 +112,8 @@ Another way to calculate binomial coefficients is as follows:
There are $n!$ permutations of $n$ elements. There are $n!$ permutations of $n$ elements.
We go through all permutations and always We go through all permutations and always
select the first $k$ elements of the permutation include the first $k$ elements of the permutation
to the subset. in the subset.
Since the order of the elements in the subset Since the order of the elements in the subset
and outside the subset does not matter, and outside the subset does not matter,
the result is divided by $k!$ and $(n-k)!$ the result is divided by $k!$ and $(n-k)!$
@ -124,10 +124,9 @@ For binomial coefficients,
\[ \[
{n \choose k} = {n \choose n-k}, {n \choose k} = {n \choose n-k},
\] \]
because we can either select $k$ because we actually divide a set of $n$ elements into
elements that belong to the subset two subsets: the first contains $k$ elements
or $n-k$ elements that and the second contains $n-k$ elements.
do not belong to the subset.
The sum of binomial coefficients is The sum of binomial coefficients is
\[ \[