Solve 12*40!div4!div36!/52!+5!div47!

$$\frac { 12 \times 40 ! \div 4 ! \div 36 ! } { 52 ! + 5 ! \div 47 ! }$$

Evaluate

$$\frac{2363557804170566002896051241748859884938497162404167680000000000}{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}\approx 1.359663788 \cdot 10^{-62}$$

Solution Steps

The factorial of 40 is 815915283247897734345611269596115894272000000000.

$$\frac{\frac{\frac{12\times 815915283247897734345611269596115894272000000000}{4!}}{36!}}{52!+\frac{5!}{47!}}$$

Multiply 12 and 815915283247897734345611269596115894272000000000 to get 9790983398974772812147335235153390731264000000000.

$$\frac{\frac{\frac{9790983398974772812147335235153390731264000000000}{4!}}{36!}}{52!+\frac{5!}{47!}}$$

The factorial of 4 is 24.

$$\frac{\frac{\frac{9790983398974772812147335235153390731264000000000}{24}}{36!}}{52!+\frac{5!}{47!}}$$

Divide 9790983398974772812147335235153390731264000000000 by 24 to get 407957641623948867172805634798057947136000000000.

$$\frac{\frac{407957641623948867172805634798057947136000000000}{36!}}{52!+\frac{5!}{47!}}$$

The factorial of 36 is 371993326789901217467999448150835200000000.

$$\frac{\frac{407957641623948867172805634798057947136000000000}{371993326789901217467999448150835200000000}}{52!+\frac{5!}{47!}}$$

Divide 407957641623948867172805634798057947136000000000 by 371993326789901217467999448150835200000000 to get 1096680.

$$\frac{1096680}{52!+\frac{5!}{47!}}$$

The factorial of 52 is 80658175170943878571660636856403766975289505440883277824000000000000.

$$\frac{1096680}{80658175170943878571660636856403766975289505440883277824000000000000+\frac{5!}{47!}}$$

The factorial of 5 is 120.

$$\frac{1096680}{80658175170943878571660636856403766975289505440883277824000000000000+\frac{120}{47!}}$$

The factorial of 47 is 258623241511168180642964355153611979969197632389120000000000.

$$\frac{1096680}{80658175170943878571660636856403766975289505440883277824000000000000+\frac{120}{258623241511168180642964355153611979969197632389120000000000}}$$

Reduce the fraction \frac{120}{258623241511168180642964355153611979969197632389120000000000} to lowest terms by extracting and canceling out 120.

$$\frac{1096680}{80658175170943878571660636856403766975289505440883277824000000000000+\frac{1}{2155193679259734838691369626280099833076646936576000000000}}$$

Convert 80658175170943878571660636856403766975289505440883277824000000000000 to fraction \frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000000}{2155193679259734838691369626280099833076646936576000000000}.

$$\frac{1096680}{\frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000000}{2155193679259734838691369626280099833076646936576000000000}+\frac{1}{2155193679259734838691369626280099833076646936576000000000}}$$

Since \frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000000}{2155193679259734838691369626280099833076646936576000000000} and \frac{1}{2155193679259734838691369626280099833076646936576000000000} have the same denominator, add them by adding their numerators.

$$\frac{1096680}{\frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000000+1}{2155193679259734838691369626280099833076646936576000000000}}$$

Add 173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000000 and 1 to get 173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001.

$$\frac{1096680}{\frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}{2155193679259734838691369626280099833076646936576000000000}}$$

Divide 1096680 by \frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}{2155193679259734838691369626280099833076646936576000000000} by multiplying 1096680 by the reciprocal of \frac{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}{2155193679259734838691369626280099833076646936576000000000}.

$$1096680\times \frac{2155193679259734838691369626280099833076646936576000000000}{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}$$

Express 1096680\times \frac{2155193679259734838691369626280099833076646936576000000000}{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001} as a single fraction.

$$\frac{1096680\times 2155193679259734838691369626280099833076646936576000000000}{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}$$

Multiply 1096680 and 2155193679259734838691369626280099833076646936576000000000 to get 2363557804170566002896051241748859884938497162404167680000000000.

$$\frac{2363557804170566002896051241748859884938497162404167680000000000}{173833989309042729678551896118308072958534510578435662655660526271392122558279469397760790904795115290624000000000000000000001}$$