Why is 0 Factorial Equal to 1?

Introduction

• Topic: Understanding why 0! = 1.
• Objective: Provide a simple proof.

Definition of Factorial

• Definition: x! (x factorial) is the product of all positive integers up to x.
• Example:
  • 5! = 5 × 4 × 3 × 2 × 1 = 120
  • 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Manipulating Factorial

• Rewriting Factorials:
  • 5! can also be expressed as 5 × 4!
  • 7! can also be expressed as 7 × 6!

Generalization of the Factorial

• General Formula:
  • x! = x × (x - 1)!

Isolating (x - 1)!

• Divide both sides by x:
  • (x!)/x = (x - 1)!
• Result: (x - 1)! = x! / x

Application for x = 1

• Plugging in x = 1:
  • 1 - 1! = 1! / 1
  • 0! = 1 / 1
• Conclusion:
  • 0! = 1

Conclusion

• The proof that 0 factorial equals 1 has been demonstrated.
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