Finite Math Examples

120 C 10

${}_{120}C_{10}$

Step 1

Evaluate

120 C 10

${}_{120}C_{10}$ using the formula

n C r = \frac{n!}{(n - r)! r!}

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

\frac{120!}{(120 - 10)! 10!}

$\frac{120!}{(120 - 10)! 10!}$

Step 2

Subtract

10

$10$ from

120

$120$ .

\frac{120!}{(110)! 10!}

$\frac{120!}{(110)! 10!}$

Step 3

Simplify

\frac{120!}{(110)! 10!}

$\frac{120!}{(110)! 10!}$ .

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Step 3.1

Rewrite

120!

$120!$ as

120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111 \cdot 110!

$120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111 \cdot 110!$ .

\frac{120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111 \cdot 110!}{(110)! 10!}

$\frac{120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111 \cdot 110!}{(110)! 10!}$

Step 3.2

Cancel the common factor of

110!

$110!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111 \cdot 110!}{(110)! 10!}$

Step 3.2.2

Rewrite the expression.

$\frac{120 \cdot 119 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$120$ by

$119$ .

$\frac{14280 \cdot 118 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.2

Multiply

$14280$ by

$118$ .

$\frac{1685040 \cdot 117 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.3

Multiply

$1685040$ by

$117$ .

$\frac{197149680 \cdot 116 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.4

Multiply

$197149680$ by

$116$ .

$\frac{22869362880 \cdot 115 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.5

Multiply

$22869362880$ by

$115$ .

$\frac{2629976731200 \cdot 114 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.6

Multiply

$2629976731200$ by

$114$ .

$\frac{299817347356800 \cdot 113 \cdot 112 \cdot 111}{10!}$

Step 3.3.7

Multiply

$299817347356800$ by

$113$ .

$\frac{33879360251318400 \cdot 112 \cdot 111}{10!}$

Step 3.3.8

Multiply

$33879360251318400$ by

$112$ .

$\frac{3794488348147660800 \cdot 111}{10!}$

Step 3.3.9

Multiply

$3794488348147660800$ by

$111$ .

$\frac{421188206644390348800}{10!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$10!$ to

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

$\frac{421188206644390348800}{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

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

$9$ .

$\frac{421188206644390348800}{90 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$90$ by

$8$ .

$\frac{421188206644390348800}{720 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$720$ by

$7$ .

$\frac{421188206644390348800}{5040 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply

$5040$ by

$6$ .

$\frac{421188206644390348800}{30240 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply

$30240$ by

$5$ .

$\frac{421188206644390348800}{151200 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.6

Multiply

$151200$ by

$4$ .

$\frac{421188206644390348800}{604800 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.7

Multiply

$604800$ by

$3$ .

$\frac{421188206644390348800}{1814400 \cdot 2 \cdot 1}$

Step 3.4.2.8

Multiply

$1814400$ by

$2$ .

$\frac{421188206644390348800}{3628800 \cdot 1}$

Step 3.4.2.9

Multiply

$3628800$ by

$1$ .

$\frac{421188206644390348800}{3628800}$

Step 3.5

Divide

$421188206644390348800$ by

$3628800$ .

$116068178638776$