## Permutations and Combinations

### 2.4 Aside: Leonhard Euler and Factorials

We have:

\(1! = 1\)

\(2! = 2 \times 1 = 2\)

\(3! = 3 \times 2 \times 1 = 6\)

\(4! = 4 \times 3 \times 2 \times 1 = 24\)

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

\(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

\(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)

\(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\)

A graph of these values appears as follows:

In 1729, at the age of 22, Swiss mathematician Leonhard Euler wondered if there is a formula for a general function that “connects the dots” of the factorial numbers. And he found one! He called it the *Gamma Function*. The curious thing is that one can input rational and irrational values into the Gamma function and obtain meaningful answers. The Gamma function gives \(1! = 1\) and \(0! = 1\) (just as the mathematics we saw earlier suggests) and it also gives:

\(\left(\dfrac{1}{2}\right)! = \dfrac{\sqrt{\pi}}{2}\).

Crazy!

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