Solve 80!/(80-71)!80!(0,7)^71(1-0,3)^19

$$\frac { 80 ! } { ( 80 - 71 ) ! 80 ! } ( 0,7 ) ^ { 71 } ( 1 - 0,3 ) ^ { 19 }$$

Evaluate

$$\frac{1635782513474434908477160959077878011007714974754996979744938053160034289607}{51840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}\approx 3,155444663 \cdot 10^{-20}$$

Solution Steps

The factorial of 80 is 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000.

$$\frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{\left(80-71\right)!\times 80!}\times 0,7^{71}\left(1-0,3\right)^{19}$$

Subtract 71 from 80 to get 9.

$$\frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{9!\times 80!}\times 0,7^{71}\left(1-0,3\right)^{19}$$

The factorial of 9 is 362880.

$$\frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{362880\times 80!}\times 0,7^{71}\left(1-0,3\right)^{19}$$

The factorial of 80 is 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000.

$$\frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{362880\times 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}\times 0,7^{71}\left(1-0,3\right)^{19}$$

Multiply 362880 and 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000 to get 25971124572948208576741209357469928792247360456397684295181475422729428005028636412027706713438861813350400000000000000000000.

$$\frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{25971124572948208576741209357469928792247360456397684295181475422729428005028636412027706713438861813350400000000000000000000}\times 0,7^{71}\left(1-0,3\right)^{19}$$

Reduce the fraction \frac{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}{25971124572948208576741209357469928792247360456397684295181475422729428005028636412027706713438861813350400000000000000000000} to lowest terms by extracting and canceling out 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000.

$$\frac{1}{362880}\times 0,7^{71}\left(1-0,3\right)^{19}$$

Calculate 0,7 to the power of 71 and get 0,00000000001004525211269079039999221534496697502180541686174722466474743.

$$\frac{1}{362880}\times 0,00000000001004525211269079039999221534496697502180541686174722466474743\left(1-0,3\right)^{19}$$

Multiply \frac{1}{362880} and 0,00000000001004525211269079039999221534496697502180541686174722466474743 to get \frac{143503601609868434285603076356671071740077383739246066639249}{5184000000000000000000000000000000000000000000000000000000000000000000000000}.

$$\frac{143503601609868434285603076356671071740077383739246066639249}{5184000000000000000000000000000000000000000000000000000000000000000000000000}\left(1-0,3\right)^{19}$$

Subtract 0,3 from 1 to get 0,7.

$$\frac{143503601609868434285603076356671071740077383739246066639249}{5184000000000000000000000000000000000000000000000000000000000000000000000000}\times 0,7^{19}$$

Calculate 0,7 to the power of 19 and get 0,0011398895185373143.

$$\frac{143503601609868434285603076356671071740077383739246066639249}{5184000000000000000000000000000000000000000000000000000000000000000000000000}\times 0,0011398895185373143$$

Multiply \frac{143503601609868434285603076356671071740077383739246066639249}{5184000000000000000000000000000000000000000000000000000000000000000000000000} and 0,0011398895185373143 to get \frac{1635782513474434908477160959077878011007714974754996979744938053160034289607}{51840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.

$$\frac{1635782513474434908477160959077878011007714974754996979744938053160034289607}{51840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

Factor

$$\frac{7 ^ {89}}{2 ^ {97} \cdot 3 ^ {4} \cdot 5 ^ {91}} = 3.1554446633380304 \times 10^{-20}$$