Finite Math Examples

${}_{100}C_{7}$

Step 1

Evaluate ${}_{100}C_{7}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{100!}{(100 - 7)! 7!}$

Step 2

Subtract $7$ from $100$ .

$\frac{100!}{(93)! 7!}$

Step 3

Simplify $\frac{100!}{(93)! 7!}$ .

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Step 3.1

Rewrite $100!$ as $100 \cdot 99 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94 \cdot 93!$ .

$\frac{100 \cdot 99 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94 \cdot 93!}{(93)! 7!}$

Step 3.2

Cancel the common factor of $93!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{100 \cdot 99 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94 \cdot 93!}{(93)! 7!}$

Step 3.2.2

Rewrite the expression.

$\frac{100 \cdot 99 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94}{7!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $100$ by $99$ .

$\frac{9900 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94}{7!}$

Step 3.3.2

Multiply $9900$ by $98$ .

$\frac{970200 \cdot 97 \cdot 96 \cdot 95 \cdot 94}{7!}$

Step 3.3.3

Multiply $970200$ by $97$ .

$\frac{94109400 \cdot 96 \cdot 95 \cdot 94}{7!}$

Step 3.3.4

Multiply $94109400$ by $96$ .

$\frac{9034502400 \cdot 95 \cdot 94}{7!}$

Step 3.3.5

Multiply $9034502400$ by $95$ .

$\frac{858277728000 \cdot 94}{7!}$

Step 3.3.6

Multiply $858277728000$ by $94$ .

$\frac{80678106432000}{7!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand $7!$ to $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{80678106432000}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

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 $6$ .

$\frac{80678106432000}{42 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply $42$ by $5$ .

$\frac{80678106432000}{210 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply $210$ by $4$ .

$\frac{80678106432000}{840 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply $840$ by $3$ .

$\frac{80678106432000}{2520 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply $2520$ by $2$ .

$\frac{80678106432000}{5040 \cdot 1}$

Step 3.4.2.6

Multiply $5040$ by $1$ .

$\frac{80678106432000}{5040}$

Step 3.5

Divide $80678106432000$ by $5040$ .

$16007560800$