Pascal's Identity

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

${n \choose k}={n-1\choose k-1}+{n-1\choose k}$

for $\{ k,n \in \bbfont{N} | k<n \}$ (Error compiling LaTeX. Unknown error_msg)

Proof

We have $\{ k,n \in \bbfont{N} | k<n \}$ (Error compiling LaTeX. Unknown error_msg):

$\binom{n-1}{k-1}+\binom{n-1}{k}=\frac{(n-1)!}{(k-1)!(n-k)!}+\frac{(n-1)!}{k!(n-k-1)!}$

$=(n-1)!\left[\frac{k}{k!(n-k)!}+\frac{n-k}{k!(n-k)!}$ (Error compiling LaTeX. Unknown error_msg)

$=(n-1)!\frac{n}{k!(n-k)!}$

$=\frac{n!}{k!(n-k)!}$

$=\binom{n}{k}$

∎

Alternate Proof

Here, we prove this using committee forming.

Consider picking one fixed object out of $n$ objects. Then, we can choose $k$ objects including that one in $\binom{n-1}{k-1}$ ways.

Because our final group of objects either contains the specified one or doesn't, we can choose the group in $\binom{n-1}{k-1}+\binom{n-1}{k}$ ways.

But we already know they can be picked in $\binom{n}{k}$ ways, so

${n \choose k}={n-1\choose k-1}+{n-1\choose k}$

∎

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.

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