Finite Math Examples

${}_{10}C_{5}$

Step 1

Evaluate

${}_{10}C_{5}$ using the formula

${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{10!}{(10 - 5)! 5!}$

Step 2

Subtract

$5$ from

$10$ .

$\frac{10!}{(5)! 5!}$

Step 3

Simplify

$\frac{10!}{(5)! 5!}$ .

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Step 3.1

Rewrite

$10!$ as

$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5!$ .

$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5!}{(5)! 5!}$

Step 3.2

Cancel the common factor of

$5!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5!}{(5)! 5!}$

Step 3.2.2

Rewrite the expression.

$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6}{5!}$

$\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6}{5!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$10$ by

$9$ .

$\frac{90 \cdot 8 \cdot 7 \cdot 6}{5!}$

Step 3.3.2

Multiply

$90$ by

$8$ .

$\frac{720 \cdot 7 \cdot 6}{5!}$

Step 3.3.3

Multiply

$720$ by

$7$ .

$\frac{5040 \cdot 6}{5!}$

Step 3.3.4

Multiply

$5040$ by

$6$ .

$\frac{30240}{5!}$

$\frac{30240}{5!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$5!$ to

$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{30240}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply

$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply

$5$ by

$4$ .

$\frac{30240}{20 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$20$ by

$3$ .

$\frac{30240}{60 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$60$ by

$2$ .

$\frac{30240}{120 \cdot 1}$

Step 3.4.2.4

Multiply

$120$ by

$1$ .

$\frac{30240}{120}$

$\frac{30240}{120}$

$\frac{30240}{120}$

Step 3.5

Divide

$30240$ by

$120$ .

$252$

$252$

${}_{10}C_{5}$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$