Hello. Here is a short guide on factorials.
Before going into the definition of factorials. Here are some examples of factorials.
\(3! = 3 \times 2 \times 1 = 1 \times 2 \times 3 = 6\)
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 1 \times 2 \times 3 \times 4 \times 5 = 120\)
As you can see, a factorial is a compact way of expressing the multiplication of descending (or ascending) positive whole numbers.
With factorials, the general case is:
\[n! = n \times (n - 1) \times (n - 2) \times ... \times 1\]
or
\[n! = 1 \times 2 \times ... \times (n -1) \times n\]
for \(n = \{2, 3, 4, 5, ...\}\).
There are also some special cases with \(1! = 1\) and \(0! = 1\).
You may encounter fractions with factorials in the numerator and in the denominator. Here are some examples.
Example One
\[\dfrac{4!}{2!} = \dfrac{4 \times 3 \times 2 \times 1}{2 \times 1} = 4 \times 3 = 12\]
Example Two
\[\dfrac{10!}{9!} = \dfrac{10 \times 9!}{9!} = 10\]
Example Three
\[\dfrac{8!}{4! \times 2!} = \dfrac{8 \times 7 \times 6 \times 5 \times 4!}{4! \times 2!} = \dfrac{8 \times 7 \times 6 \times 5 }{2} = 7 \times 6 \times 5 \times 4 = 840\]
Example Four
\[\dfrac{4! \times 5!}{6!} = \dfrac{4! \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4!} = \dfrac{4 \times 3 \times 2 \times 1}{6} = \dfrac{24}{6} = 4\]