Finite Math Examples

${}_{20}C_{10}$

Step 1

Evaluate

${}_{20}C_{10}$ using the formula

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

$\frac{20!}{(20 - 10)! 10!}$

Step 2

Subtract

$10$ from

$20$ .

$\frac{20!}{(10)! 10!}$

Step 3

Simplify

$\frac{20!}{(10)! 10!}$ .

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Step 3.1

Rewrite

$20!$ as

$20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10!$ .

$\frac{20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10!}{(10)! 10!}$

Step 3.2

Cancel the common factor of

$10!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10!}{(10)! 10!}$

Step 3.2.2

Rewrite the expression.

$\frac{20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$20$ by

$19$ .

$\frac{380 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.2

Multiply

$380$ by

$18$ .

$\frac{6840 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.3

Multiply

$6840$ by

$17$ .

$\frac{116280 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.4

Multiply

$116280$ by

$16$ .

$\frac{1860480 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.5

Multiply

$1860480$ by

$15$ .

$\frac{27907200 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.6

Multiply

$27907200$ by

$14$ .

$\frac{390700800 \cdot 13 \cdot 12 \cdot 11}{10!}$

Step 3.3.7

Multiply

$390700800$ by

$13$ .

$\frac{5079110400 \cdot 12 \cdot 11}{10!}$

Step 3.3.8

Multiply

$5079110400$ by

$12$ .

$\frac{60949324800 \cdot 11}{10!}$

Step 3.3.9

Multiply

$60949324800$ by

$11$ .

$\frac{670442572800}{10!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$10!$ to

$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{670442572800}{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply

$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply

$10$ by

$9$ .

$\frac{670442572800}{90 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$90$ by

$8$ .

$\frac{670442572800}{720 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$720$ by

$7$ .

$\frac{670442572800}{5040 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply

$5040$ by

$6$ .

$\frac{670442572800}{30240 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply

$30240$ by

$5$ .

$\frac{670442572800}{151200 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.6

Multiply

$151200$ by

$4$ .

$\frac{670442572800}{604800 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.7

Multiply

$604800$ by

$3$ .

$\frac{670442572800}{1814400 \cdot 2 \cdot 1}$

Step 3.4.2.8

Multiply

$1814400$ by

$2$ .

$\frac{670442572800}{3628800 \cdot 1}$

Step 3.4.2.9

Multiply

$3628800$ by

$1$ .

$\frac{670442572800}{3628800}$

Step 3.5

Divide

$670442572800$ by

$3628800$ .

$184756$