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# 0!

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### 0!

1. 1. Why 0!=1Example (1)If n! is defined as the product of all positive integers from 1 to n, then: 1! = 1×1 = 1 2! = 1×2 = 2 3! = 1×2×3 = 6 4! = 1 × 2 × 3 × 4 = 24 ... 𝑛! = 1 × 2 × 3 ×. . .× (𝑛 − 2) × (𝑛 − 1) × 𝑛 and so on. Logically, n! can also be expressed 𝑛 × (𝑛 − 1)! . Therefore, at n=1, using 𝑛! = 𝑛 × (𝑛 − 1)! 1! = 1 × 0! which simplifies to 1 = 0!
2. 2. Example (2) Why 0!=1The idea of the factorial (in simple terms) is used to compute the number of permutations ofarranging a set of n numbers. n Number of Permutations (n!) Visual example 1 1 {1} 2 2 {1,2}, {2,1} 3 6 {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}Therefore, 0 1 {} It can be said that an empty set can only be ordered one way, so 0! = 1.