Finite Math Examples

${}_{52}C_{7}$

Step 1

Evaluate

${}_{52}C_{7}$ using the formula

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

$\frac{52!}{(52 - 7)! 7!}$

Step 2

Subtract

$7$ from

$52$ .

$\frac{52!}{(45)! 7!}$

Step 3

Simplify

$\frac{52!}{(45)! 7!}$ .

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Step 3.1

Rewrite

$52!$ as

$52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45!$ .

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45!}{(45)! 7!}$

Step 3.2

Cancel the common factor of

$45!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45!}{(45)! 7!}$

Step 3.2.2

Rewrite the expression.

$\frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46}{7!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$52$ by

$51$ .

$\frac{2652 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46}{7!}$

Step 3.3.2

Multiply

$2652$ by

$50$ .

$\frac{132600 \cdot 49 \cdot 48 \cdot 47 \cdot 46}{7!}$

Step 3.3.3

Multiply

$132600$ by

$49$ .

$\frac{6497400 \cdot 48 \cdot 47 \cdot 46}{7!}$

Step 3.3.4

Multiply

$6497400$ by

$48$ .

$\frac{311875200 \cdot 47 \cdot 46}{7!}$

Step 3.3.5

Multiply

$311875200$ by

$47$ .

$\frac{14658134400 \cdot 46}{7!}$

Step 3.3.6

Multiply

$14658134400$ by

$46$ .

$\frac{674274182400}{7!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$7!$ to

$7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{674274182400}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply

$7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply

$7$ by

$6$ .

$\frac{674274182400}{42 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$42$ by

$5$ .

$\frac{674274182400}{210 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$210$ by

$4$ .

$\frac{674274182400}{840 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply

$840$ by

$3$ .

$\frac{674274182400}{2520 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply

$2520$ by

$2$ .

$\frac{674274182400}{5040 \cdot 1}$

Step 3.4.2.6

Multiply

$5040$ by

$1$ .

$\frac{674274182400}{5040}$

Step 3.5

Divide

$674274182400$ by

$5040$ .

$133784560$