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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pascal's rule ： ウィキペディア英語版 Pascal's rule In mathematics, Pascal's rule is a combinatorial identity about binomial coefficients. It states that for any natural number ''n'' we have :$+\; =\; \backslash quad\backslash text1\; \backslash le\; k\; \backslash le\; n$ where $$ is a binomial coefficient. This is also commonly written :$$ + = \quad\text 1 \le k \le n + 1 ==Combinatorial proof== Pascal's rule has an intuitive combinatorial meaning. Recall that $$ counts in how many ways can we pick a subset with ''b'' elements out from a set with ''a'' elements. Therefore, the right side of the identity $$ is counting how many ways can we get a ''k''-subset out from a set with ''n'' elements. Now, suppose you distinguish a particular element 'X' from the set with ''n'' elements. Thus, every time you choose ''k'' elements to form a subset there are two possibilities: ''X'' belongs to the chosen subset or not. If ''X'' is in the subset, you only really need to choose ''k'' − 1 more objects (since it is known that ''X'' will be in the subset) out from the remaining ''n'' − 1 objects. This can be accomplished in $n-1\backslash choose\; k-1$ ways. When ''X'' is not in the subset, you need to choose all the ''k'' elements in the subset from the ''n'' − 1 objects that are not ''X''. This can be done in $n-1\backslash choose\; k$ ways. We conclude that the numbers of ways to get a ''k''-subset from the ''n''-set, which we know is $$, is also the number $+\; .$ See also Bijective proof. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pascal's rule」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.