Finite Math Examples

${}_{67}C_{6}$

Step 1

Evaluate ${}_{67}C_{6}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{67!}{(67 - 6)! 6!}$

Step 2

Subtract $6$ from $67$ .

$\frac{67!}{(61)! 6!}$

Step 3

Simplify $\frac{67!}{(61)! 6!}$ .

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Step 3.1

Rewrite $67!$ as $67 \cdot 66 \cdot 65 \cdot 64 \cdot 63 \cdot 62 \cdot 61!$ .

$\frac{67 \cdot 66 \cdot 65 \cdot 64 \cdot 63 \cdot 62 \cdot 61!}{(61)! 6!}$

Step 3.2

Cancel the common factor of $61!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{67 \cdot 66 \cdot 65 \cdot 64 \cdot 63 \cdot 62 \cdot 61!}{(61)! 6!}$

Step 3.2.2

Rewrite the expression.

$\frac{67 \cdot 66 \cdot 65 \cdot 64 \cdot 63 \cdot 62}{6!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $67$ by $66$ .

$\frac{4422 \cdot 65 \cdot 64 \cdot 63 \cdot 62}{6!}$

Step 3.3.2

Multiply $4422$ by $65$ .

$\frac{287430 \cdot 64 \cdot 63 \cdot 62}{6!}$

Step 3.3.3

Multiply $287430$ by $64$ .

$\frac{18395520 \cdot 63 \cdot 62}{6!}$

Step 3.3.4

Multiply $18395520$ by $63$ .

$\frac{1158917760 \cdot 62}{6!}$

Step 3.3.5

Multiply $1158917760$ by $62$ .

$\frac{71852901120}{6!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand $6!$ to $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{71852901120}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply $6$ by $5$ .

$\frac{71852901120}{30 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply $30$ by $4$ .

$\frac{71852901120}{120 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply $120$ by $3$ .

$\frac{71852901120}{360 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply $360$ by $2$ .

$\frac{71852901120}{720 \cdot 1}$

Step 3.4.2.5

Multiply $720$ by $1$ .

$\frac{71852901120}{720}$

Step 3.5

Divide $71852901120$ by $720$ .

$99795696$