Solve 40!div0!div40!*(12!div5!div7!)/52!div5!div47!

$$\frac{ 40 ! \div 0 ! \div 40 ! \times (12 ! \div 5 ! \div 7 ! ) }{ 52 ! \div 5 ! \div 47 ! }$$

Evaluate

$$\frac{33}{108290}\approx 0.000304737$$

Solution Steps

The factorial of 40 is 815915283247897734345611269596115894272000000000.

$$\frac{\frac{\frac{815915283247897734345611269596115894272000000000}{0!}}{40!}\times \frac{\frac{12!}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 0 is 1.

$$\frac{\frac{\frac{815915283247897734345611269596115894272000000000}{1}}{40!}\times \frac{\frac{12!}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

Anything divided by one gives itself.

$$\frac{\frac{815915283247897734345611269596115894272000000000}{40!}\times \frac{\frac{12!}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 40 is 815915283247897734345611269596115894272000000000.

$$\frac{\frac{815915283247897734345611269596115894272000000000}{815915283247897734345611269596115894272000000000}\times \frac{\frac{12!}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

Divide 815915283247897734345611269596115894272000000000 by 815915283247897734345611269596115894272000000000 to get 1.

$$\frac{1\times \frac{\frac{12!}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 12 is 479001600.

$$\frac{1\times \frac{\frac{479001600}{5!}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 5 is 120.

$$\frac{1\times \frac{\frac{479001600}{120}}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

Divide 479001600 by 120 to get 3991680.

$$\frac{1\times \frac{3991680}{7!}}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 7 is 5040.

$$\frac{1\times \frac{3991680}{5040}}{\frac{\frac{52!}{5!}}{47!}}$$

Divide 3991680 by 5040 to get 792.

$$\frac{1\times 792}{\frac{\frac{52!}{5!}}{47!}}$$

Multiply 1 and 792 to get 792.

$$\frac{792}{\frac{\frac{52!}{5!}}{47!}}$$

The factorial of 52 is 80658175170943878571660636856403766975289505440883277824000000000000.

$$\frac{792}{\frac{\frac{80658175170943878571660636856403766975289505440883277824000000000000}{5!}}{47!}}$$

The factorial of 5 is 120.

$$\frac{792}{\frac{\frac{80658175170943878571660636856403766975289505440883277824000000000000}{120}}{47!}}$$

Divide 80658175170943878571660636856403766975289505440883277824000000000000 by 120 to get 672151459757865654763838640470031391460745878674027315200000000000.

$$\frac{792}{\frac{672151459757865654763838640470031391460745878674027315200000000000}{47!}}$$

The factorial of 47 is 258623241511168180642964355153611979969197632389120000000000.

$$\frac{792}{\frac{672151459757865654763838640470031391460745878674027315200000000000}{258623241511168180642964355153611979969197632389120000000000}}$$

Divide 672151459757865654763838640470031391460745878674027315200000000000 by 258623241511168180642964355153611979969197632389120000000000 to get 2598960.

$$\frac{792}{2598960}$$

Reduce the fraction \frac{792}{2598960} to lowest terms by extracting and canceling out 24.

$$\frac{33}{108290}$$

Factor

$$\frac{3 \cdot 11}{2 \cdot 5 \cdot 7 ^ {2} \cdot 13 \cdot 17} = 0.0003047372795271955$$