Finite Math Examples

${}_{25}C_{7}$

Step 1

Evaluate

${}_{25}C_{7}$ using the formula

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

$\frac{25!}{(25 - 7)! 7!}$

Step 2

Subtract

$7$ from

$25$ .

$\frac{25!}{(18)! 7!}$

Step 3

Simplify

$\frac{25!}{(18)! 7!}$ .

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Step 3.1

Rewrite

$25!$ as

$25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18!$ .

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18!}{(18)! 7!}$

Step 3.2

Cancel the common factor of

$18!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18!}{(18)! 7!}$

Step 3.2.2

Rewrite the expression.

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19}{7!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply

$25$ by

$24$ .

$\frac{600 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19}{7!}$

Step 3.3.2

Multiply

$600$ by

$23$ .

$\frac{13800 \cdot 22 \cdot 21 \cdot 20 \cdot 19}{7!}$

Step 3.3.3

Multiply

$13800$ by

$22$ .

$\frac{303600 \cdot 21 \cdot 20 \cdot 19}{7!}$

Step 3.3.4

Multiply

$303600$ by

$21$ .

$\frac{6375600 \cdot 20 \cdot 19}{7!}$

Step 3.3.5

Multiply

$6375600$ by

$20$ .

$\frac{127512000 \cdot 19}{7!}$

Step 3.3.6

Multiply

$127512000$ by

$19$ .

$\frac{2422728000}{7!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand

$7!$ to

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

$\frac{2422728000}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

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

$6$ .

$\frac{2422728000}{42 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply

$42$ by

$5$ .

$\frac{2422728000}{210 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply

$210$ by

$4$ .

$\frac{2422728000}{840 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply

$840$ by

$3$ .

$\frac{2422728000}{2520 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply

$2520$ by

$2$ .

$\frac{2422728000}{5040 \cdot 1}$

Step 3.4.2.6

Multiply

$5040$ by

$1$ .

$\frac{2422728000}{5040}$

Step 3.5

Divide

$2422728000$ by

$5040$ .

$480700$