1 Permutations And Combinations: Proof by Induction
We have the relation expressed by the arrows can be written mathematically as:
1
1
n
n
n
k
k
k
We could also derive this formula by considering the definition of “n choose k” as the
number of subsets.
When we have n+1 items to choose from, we can pick out one of the items, call it A, and
then say: every subset having exactly k items either does contain A, or does not contain
A. If it does not contain A, then it is one of the subsets that contains k items, and is drawn
from the set of exactly n items that we have when we do not contain A. There are exactly
n
k
of these subsets. OR, it does contain the element A.
It must also have
1
k
other
elements, and they must be drawn from the
1
n
other elements in the set. So the number
of such subsets (the ones that do contain A) is exactly
1
n
k
. The subsets that we count
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 Fall '09
 Boros
 Natural number

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