Solve =4!/1!3!*7/3!*4!/frac{11!{4!7!}}

$$= \frac { \frac { 4 ! } { 1 ! 3 ! } \cdot \frac { 7 } { 3 ! \cdot 4 ! } } { \frac { 11 ! } { 4 ! 7 ! } }$$

Evaluate

$$\frac{7}{11880}\approx 0.000589226$$

Solution Steps

The factorial of 4 is 24.

$$\frac{\frac{24}{1!\times 3!}\times \frac{7}{3!\times 4!}}{\frac{11!}{4!\times 7!}}$$

The factorial of 1 is 1.

$$\frac{\frac{24}{1\times 3!}\times \frac{7}{3!\times 4!}}{\frac{11!}{4!\times 7!}}$$

The factorial of 3 is 6.

$$\frac{\frac{24}{1\times 6}\times \frac{7}{3!\times 4!}}{\frac{11!}{4!\times 7!}}$$

Multiply 1 and 6 to get 6.

$$\frac{\frac{24}{6}\times \frac{7}{3!\times 4!}}{\frac{11!}{4!\times 7!}}$$

Divide 24 by 6 to get 4.

$$\frac{4\times \frac{7}{3!\times 4!}}{\frac{11!}{4!\times 7!}}$$

The factorial of 3 is 6.

$$\frac{4\times \frac{7}{6\times 4!}}{\frac{11!}{4!\times 7!}}$$

The factorial of 4 is 24.

$$\frac{4\times \frac{7}{6\times 24}}{\frac{11!}{4!\times 7!}}$$

Multiply 6 and 24 to get 144.

$$\frac{4\times \frac{7}{144}}{\frac{11!}{4!\times 7!}}$$

Express 4\times \frac{7}{144} as a single fraction.

$$\frac{\frac{4\times 7}{144}}{\frac{11!}{4!\times 7!}}$$

Multiply 4 and 7 to get 28.

$$\frac{\frac{28}{144}}{\frac{11!}{4!\times 7!}}$$

Reduce the fraction \frac{28}{144} to lowest terms by extracting and canceling out 4.

$$\frac{\frac{7}{36}}{\frac{11!}{4!\times 7!}}$$

The factorial of 11 is 39916800.

$$\frac{\frac{7}{36}}{\frac{39916800}{4!\times 7!}}$$

The factorial of 4 is 24.

$$\frac{\frac{7}{36}}{\frac{39916800}{24\times 7!}}$$

The factorial of 7 is 5040.

$$\frac{\frac{7}{36}}{\frac{39916800}{24\times 5040}}$$

Multiply 24 and 5040 to get 120960.

$$\frac{\frac{7}{36}}{\frac{39916800}{120960}}$$

Divide 39916800 by 120960 to get 330.

$$\frac{\frac{7}{36}}{330}$$

Express \frac{\frac{7}{36}}{330} as a single fraction.

$$\frac{7}{36\times 330}$$

Multiply 36 and 330 to get 11880.

$$\frac{7}{11880}$$

Factor

$$\frac{7}{2 ^ {3} \cdot 3 ^ {3} \cdot 5 \cdot 11} = 0.0005892255892255892$$