Difference between revisions of "Pascal's Identity"
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<math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math> | <math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math> | ||
| − | for <math>\{ k,n \in \bbfont{N} | k<n \}</math> | + | for <math>\{ k,n \in \bbfont{N} | k<n \}</math>. |
| + | |||
| + | This can also be read as that the number of ways to choose <math>k</math> things from <math>n</math> things is equal to the number of ways to choose <math>k-1</math> things from <math>n-1</math> things added to the number of ways to choose <math>k</math> things from <math>n-1</math> things. | ||
| + | |||
| + | Technically, [[combinations]] can also be applied to non-integer values of <math>k</math>, in which case the identity still holds. | ||
== Proof == | == Proof == | ||
Revision as of 17:12, 9 December 2007
Pascal's identity is a common and useful theorem in the realm of combinatorics dealing with combinations (also known as binomial coefficients), and is often used to reduce large, complicated combinations.
Pascal's identity is also known as Pascal's rule, Pascal's formula, and occasionally Pascal's theorem.
Theorem
Pascal's identity states that
for $\{ k,n \in \bbfont{N} | k<n \}$ (Error compiling LaTeX. Unknown error_msg).
This can also be read as that the number of ways to choose 
things from 
things is equal to the number of ways to choose 
things from 
things added to the number of ways to choose things from things.
Technically, combinations can also be applied to non-integer values of , in which case the identity still holds.
Proof
We have $\{ k,n \in \bbfont{N} | k<n \}$ (Error compiling LaTeX. Unknown error_msg):
$=(n-1)!\left[\frac{k}{k!(n-k)!}+\frac{n-k}{k!(n-k)!}$ (Error compiling LaTeX. Unknown error_msg)
∎
Alternate Proof
Here, we prove this using committee forming.
Consider picking one fixed object out of objects. Then, we can choose objects including that one in 
ways.
Because our final group of objects either contains the specified one or doesn't, we can choose the group in 
ways.
But we already know they can be picked in 
ways, so
![\[{n \choose k}={n-1\choose k-1}+{n-1\choose k}\]](http://latex.artofproblemsolving.com/b/9/e/b9e814ea2d14ccaf430a823f7e86364bb1018a6b.png)
∎
History
Pascal's identity was probably first derived by Blaise Pascal, a 19th century French mathematician, whom the theorem is named after. Pascal also did extensive other work on combinatorics, which is why Pascal's Triangle is named after him.
