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单词 ProofOfPascalsRule
释义

proof of Pascal’s rule


The definition of (nk) is the number of k-subsets out from an n-set. Using this combinatorial meaning the proof is straightforward.

Let x a distinct element from the n-set. As previously stated, (nk) counts the number of subsets with k elements, chosen from the set with n elements. Now, some of these subsets will contain x and some others don’t.

The number of k-subsets not containing x is (n-1k), since we need to choose k elements from the n-1 elements different from x.

The number of k-subsets containing x is (n-1k-1), because if it is given that x is in the subset, we only need to choose the remaining k-1 elements from the n-1 elements that are different from x.

Thus

(nk)=(n-1k)+(n-1k-1).
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更新时间:2025/5/4 2:50:52