Finite Math Examples

52 C 5

${}_{52}C_{5}$

Step 1

Evaluate

52 C 5

${}_{52}C_{5}$ using the formula

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

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

\frac{52!}{(52 - 5)! 5!}

$\frac{52!}{(52 - 5)! 5!}$

Step 2

Subtract

5

$5$ from

52

$52$ .

\frac{52!}{(47)! 5!}

$\frac{52!}{(47)! 5!}$

Step 3

Simplify

\frac{52!}{(47)! 5!}

$\frac{52!}{(47)! 5!}$ .

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Step 3.1

Rewrite

52!

$52!$ as

52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!

$52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!$ .

\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!}{(47)! 5!}

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!}{(47)! 5!}$

Step 3.2

Cancel the common factor of

47!

$47!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!}{(47)! 5!}$

Step 3.2.2

Rewrite the expression.

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48}{5!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$52$ by

$51$ .

$\frac{2652 \cdot 50 \cdot 49 \cdot 48}{5!}$

Step 3.3.2

Multiply

$2652$ by

$50$ .

$\frac{132600 \cdot 49 \cdot 48}{5!}$

Step 3.3.3

Multiply

$132600$ by

$49$ .

$\frac{6497400 \cdot 48}{5!}$

Step 3.3.4

Multiply

$6497400$ by

$48$ .

$\frac{311875200}{5!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$5!$ to

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

$\frac{311875200}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

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

$4$ .

$\frac{311875200}{20 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$20$ by

$3$ .

$\frac{311875200}{60 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$60$ by

$2$ .

$\frac{311875200}{120 \cdot 1}$

Step 3.4.2.4

Multiply

$120$ by

$1$ .

$\frac{311875200}{120}$

Step 3.5

Divide

$311875200$ by

$120$ .

$2598960$