Facts 4

Number Theory Level 1

0 ! 0 ! = ? \large \dfrac{0!}{0!} =\, ?

Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .

0 ∞ \infty 1 not defined

Ayush G Rai by Ayush G Rai , 20, India

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3 solutions

Ashish Menon
Jun 24, 2016

0 ! 0 ! = 1 1 = 1 \dfrac{0!}{0!} = \dfrac{1}{1} = \color{#3D99F6}{\boxed{1}} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Just in case if people ask why 0 ! = 1 ( n − 1 ) ! = Γ ( n ) = ∫ 0 ∞ e − x x n − 1 ⋅ d x Put n = 1 0 ! = Γ ( 1 ) = ∫ 0 ∞ e − x ⋅ 1 ⋅ d x 0 ! = 1 \text{Just in case if people ask why } 0! =1 \\ (n-1)! = \Gamma(n) = \displaystyle\int_{0}^{\infty} e^{-x}x^{n-1} \cdot dx \\ \text{Put } n =1 \\ 0! = \Gamma(1) = \displaystyle\int_{0}^{\infty} e^{-x}\cdot1 \cdot dx \\ \boxed{0! = 1}

Sabhrant Sachan - 4 years, 11 months ago

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Yup, perfec.

Ashish Menon - 4 years, 11 months ago
Viki Zeta
Jun 24, 2016

n ! = ( n + 1 ) ! ( n + 1 ) = > 0 ! = ( 0 + 1 ) ! 0 + 1 = > 0 ! = 1 ! 1 = > 0 ! = 1 So, 0 ! 0 ! = 1 1 = 1 n! = \frac{(n+1)!}{(n+1)}\\ => 0! = \frac{(0+1)!}{0+1} \\ =>0! = \frac{1!}{1} \\ => 0! = 1 \\\\ \text{So, }\frac{0!}{0!} = \frac{1}{1} = 1

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Prince Loomba
Jun 23, 2016

0! is 1 so the value is 1/1=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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