By Jacques Ozanam
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Additional info for Recreations in mathematics and natural philosophy
8! 8! = = 70 (8 − 4)! 4! 4! 4! 8! 8! = = 56 (8 − 5)! 5! 3! 5! 1, 8, 28, 56, 70, 56, 28... The next two numbers are 8 and 1. Note that there is a simple geometric way to generate this same sequence. It's called Pascal's triangle. Study the triangle illustrated on the following page. Each number on the inside of this triangle comes from adding the two numbers above it. If you happen to know about algebra, yet another way to make this triangle is to foil out (x + y)N.
Which will be explained shortly). For example, 4! means 4 times 3 times 2 times 1, which equals 4 × 3 × 2 × 1 = 24. As another example, 3! = 3 × 2 × 1 = 6. Observe that 4! = 4 × 3! (since 24 = 4 × 6). Note that 0! is defined to equal 1. The reason that 0! = 1 is so that 1! can follow the rule N! = N (N – 1)! for all positive 12 Factorials integers (N > 0). This way, 1! = 1 (1 – 1)! = 0! since 1! and 0! both equal 1. Example 1. This sequence is made from factorials. For example, 0! = 1, 1! = 1, 2!
If you happen to know about algebra, yet another way to make this triangle is to foil out (x + y)N.