Is $10^{100}$ (Googol) bigger than $100!$?

Question

Is $10^{100}$ (Googol) bigger than $100!$?

If $10^{100}$ is called as Googol, does $100!$ have any special name to be called, apart from being called as "100 factorial"?

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I ask this question because I get to know about the number $10^{100}$ on how big it is more often than $100!$. If $100!$ is bigger than $10^{100}$, then why don't we give more focus to $100!$ than the other number? Because for me, $100!$ looks simple.

Answer 1

With simple ineqalities we have:

$100!\geq 90^{10}\cdot 80^{10}\cdots 20^{10}\cdot 10^{10}$

$100!\geq (9\cdot 8 \cdots 2 \cdot 1)^{10}\cdot 10^{90}$

$100!\geq (9!)^{10}\cdot 10^{90}>10^{100}$

Answer 2

Before there was an error on the algebra, as pointed out in the comments. I try to fix the error following the same approach:

$100!=(1\times..\times 10)\times(11\times..\times 20)\times...\times(91\times..\times 100)=A_1...A_{10}$

so we estimate $A_i \ge 10^{10}$ for $i=2,..9$.

Instead we write $A_1A_{10}=(1\times 100)\times(2\times 99)\times(3 \times 97)\times...\times(10 \times 91)\ge (10^2)^{10}$.

Combining: $100!\ge (10^{10})^8 \times (10^2)^{10}=(10^{10})^{10}=10^{100}$.

Answer 3

Using $$n!>\bigg(\frac{n}{3}\bigg)^{n}, n>8$$

$$100!>\bigg(\frac{100}{3}\bigg)^{100}>10^{100}$$