Solve =[10!/4!*6!+10!/3!*7!+10!/2!*8!+10!/1!*9!+frac{10!}{10!}]frac{1}{2^10}

$$= [ \frac { 10 ! } { 4 ! \times 6 ! } + \frac { 10 ! } { 3 ! \times 7 ! } + \frac { 10 ! } { 2 ! \times 8 ! } + \frac { 10 ! } { 1 ! \times 9 ! } + \frac { 10 ! } { 10 ! } ] \frac { 1 } { 2 ^ { 10 } }$$

Evaluate

$$\frac{193}{512}=0.376953125$$

Solution Steps

Divide 10! by 10! to get 1.

$$\left(\frac{10!}{4!\times 6!}+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 10 is 3628800.

$$\left(\frac{3628800}{4!\times 6!}+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 4 is 24.

$$\left(\frac{3628800}{24\times 6!}+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 6 is 720.

$$\left(\frac{3628800}{24\times 720}+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Multiply 24 and 720 to get 17280.

$$\left(\frac{3628800}{17280}+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Divide 3628800 by 17280 to get 210.

$$\left(210+\frac{10!}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 10 is 3628800.

$$\left(210+\frac{3628800}{3!\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 3 is 6.

$$\left(210+\frac{3628800}{6\times 7!}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 7 is 5040.

$$\left(210+\frac{3628800}{6\times 5040}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Multiply 6 and 5040 to get 30240.

$$\left(210+\frac{3628800}{30240}+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Divide 3628800 by 30240 to get 120.

$$\left(210+120+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Add 210 and 120 to get 330.

$$\left(330+\frac{10!}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 10 is 3628800.

$$\left(330+\frac{3628800}{2!\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 2 is 2.

$$\left(330+\frac{3628800}{2\times 8!}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 8 is 40320.

$$\left(330+\frac{3628800}{2\times 40320}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Multiply 2 and 40320 to get 80640.

$$\left(330+\frac{3628800}{80640}+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Divide 3628800 by 80640 to get 45.

$$\left(330+45+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

Add 330 and 45 to get 375.

$$\left(375+\frac{10!}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 10 is 3628800.

$$\left(375+\frac{3628800}{1!\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 1 is 1.

$$\left(375+\frac{3628800}{1\times 9!}+1\right)\times \frac{1}{2^{10}}$$

The factorial of 9 is 362880.

$$\left(375+\frac{3628800}{1\times 362880}+1\right)\times \frac{1}{2^{10}}$$

Multiply 1 and 362880 to get 362880.

$$\left(375+\frac{3628800}{362880}+1\right)\times \frac{1}{2^{10}}$$

Divide 3628800 by 362880 to get 10.

$$\left(375+10+1\right)\times \frac{1}{2^{10}}$$

Add 375 and 10 to get 385.

$$\left(385+1\right)\times \frac{1}{2^{10}}$$

Add 385 and 1 to get 386.

$$386\times \frac{1}{2^{10}}$$

Calculate 2 to the power of 10 and get 1024.

$$386\times \frac{1}{1024}$$

Multiply 386 and \frac{1}{1024} to get \frac{386}{1024}.

$$\frac{386}{1024}$$

Reduce the fraction \frac{386}{1024} to lowest terms by extracting and canceling out 2.

$$\frac{193}{512}$$

Factor

$$\frac{193}{2 ^ {9}} = 0.376953125$$