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Math 2.0

Combinatorics Basic counting Theorem

Pascal's rule

Builds on Binomial coefficient

Statement

Let $n$ and $k$ be natural numbers. Then $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$

Remarks

For $n = 4$ and $k = 1$ it gives $\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10$ . Combinatorial proof: fix one element $x$ of an $(n+1)$ -element set. A $(k+1)$ -subset contains $x$ , and then it has $k$ other elements, or it omits $x$ . The rule builds Pascal's triangle row by row.

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To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Binomial coefficient definition
Pascal's rule statement

Further reading

Web https://en.wikipedia.org/wiki/Pascal's_rule, lead section

About this page

Tag	`0YRG241HQQ`
URI	`m2://0YRG241HQQ`
Kind	statement, theorem
Writing	W1 stated
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 0YRG241HQQ`.
Uses	Binomial coefficient
Same as	Wikidata Q17099544 Wikipedia Pascal's rule

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