Solve 330!/20!/10!+(3frac{60!/10!/50!-2frac{frac{30!}{20!}}{10!})}{frac{frac{90!}{10!}}{80!}}

$$\frac{ 3 \frac{ \frac{ 30 ! }{ 20 ! } }{ 10 ! } +(3 \frac{ \frac{ 60 ! }{ 10 ! } }{ 50 ! } -2 \frac{ \frac{ 30 ! }{ 20 ! } }{ 10 ! } ) }{ \frac{ \frac{ 90 ! }{ 10 ! } }{ 80 ! } }$$

Evaluate

$$\frac{78792103}{1992561993}\approx 0.039543112$$

Solution Steps

The factorial of 30 is 265252859812191058636308480000000.

$$\frac{3\times \frac{\frac{265252859812191058636308480000000}{20!}}{10!}+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 20 is 2432902008176640000.

$$\frac{3\times \frac{\frac{265252859812191058636308480000000}{2432902008176640000}}{10!}+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 265252859812191058636308480000000 by 2432902008176640000 to get 109027350432000.

$$\frac{3\times \frac{109027350432000}{10!}+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 10 is 3628800.

$$\frac{3\times \frac{109027350432000}{3628800}+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 109027350432000 by 3628800 to get 30045015.

$$\frac{3\times 30045015+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Multiply 3 and 30045015 to get 90135045.

$$\frac{90135045+3\times \frac{\frac{60!}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 60 is 8320987112741390144276341183223364380754172606361245952449277696409600000000000000.

$$\frac{90135045+3\times \frac{\frac{8320987112741390144276341183223364380754172606361245952449277696409600000000000000}{10!}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 10 is 3628800.

$$\frac{90135045+3\times \frac{\frac{8320987112741390144276341183223364380754172606361245952449277696409600000000000000}{3628800}}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 8320987112741390144276341183223364380754172606361245952449277696409600000000000000 by 3628800 to get 2293040981244871622651108130297443887994425872564276331693473792000000000000.

$$\frac{90135045+3\times \frac{2293040981244871622651108130297443887994425872564276331693473792000000000000}{50!}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 50 is 30414093201713378043612608166064768844377641568960512000000000000.

$$\frac{90135045+3\times \frac{2293040981244871622651108130297443887994425872564276331693473792000000000000}{30414093201713378043612608166064768844377641568960512000000000000}-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 2293040981244871622651108130297443887994425872564276331693473792000000000000 by 30414093201713378043612608166064768844377641568960512000000000000 to get 75394027566.

$$\frac{90135045+3\times 75394027566-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Multiply 3 and 75394027566 to get 226182082698.

$$\frac{90135045+226182082698-2\times \frac{\frac{30!}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 30 is 265252859812191058636308480000000.

$$\frac{90135045+226182082698-2\times \frac{\frac{265252859812191058636308480000000}{20!}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 20 is 2432902008176640000.

$$\frac{90135045+226182082698-2\times \frac{\frac{265252859812191058636308480000000}{2432902008176640000}}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 265252859812191058636308480000000 by 2432902008176640000 to get 109027350432000.

$$\frac{90135045+226182082698-2\times \frac{109027350432000}{10!}}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 10 is 3628800.

$$\frac{90135045+226182082698-2\times \frac{109027350432000}{3628800}}{\frac{\frac{90!}{10!}}{80!}}$$

Divide 109027350432000 by 3628800 to get 30045015.

$$\frac{90135045+226182082698-2\times 30045015}{\frac{\frac{90!}{10!}}{80!}}$$

Multiply 2 and 30045015 to get 60090030.

$$\frac{90135045+226182082698-60090030}{\frac{\frac{90!}{10!}}{80!}}$$

Subtract 60090030 from 226182082698 to get 226121992668.

$$\frac{90135045+226121992668}{\frac{\frac{90!}{10!}}{80!}}$$

Add 90135045 and 226121992668 to get 226212127713.

$$\frac{226212127713}{\frac{\frac{90!}{10!}}{80!}}$$

The factorial of 90 is 1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000.

$$\frac{226212127713}{\frac{\frac{1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000}{10!}}{80!}}$$

The factorial of 10 is 3628800.

$$\frac{226212127713}{\frac{\frac{1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000}{3628800}}{80!}}$$

Divide 1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000 by 3628800 to get 409423491093959848244467243612440955105150452294061057909561525491894122876451134409720053838978191039907747594240000000000000000000.

$$\frac{226212127713}{\frac{409423491093959848244467243612440955105150452294061057909561525491894122876451134409720053838978191039907747594240000000000000000000}{80!}}$$

The factorial of 80 is 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000.

$$\frac{226212127713}{\frac{409423491093959848244467243612440955105150452294061057909561525491894122876451134409720053838978191039907747594240000000000000000000}{71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000}}$$

Divide 409423491093959848244467243612440955105150452294061057909561525491894122876451134409720053838978191039907747594240000000000000000000 by 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000 to get 5720645481903.

$$\frac{226212127713}{5720645481903}$$

Reduce the fraction \frac{226212127713}{5720645481903} to lowest terms by extracting and canceling out 2871.

$$\frac{78792103}{1992561993}$$

Factor

$$\frac{13 \cdot 167 \cdot 36293}{3 ^ {2} \cdot 17 \cdot 41 \cdot 43 \cdot 83 \cdot 89} = 0.03954311247369055$$