Finite Math Examples

${}_{42}C_{10}$

Step 1

Evaluate ${}_{42}C_{10}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{42!}{(42 - 10)! 10!}$

Step 2

Subtract $10$ from $42$ .

$\frac{42!}{(32)! 10!}$

Step 3

Simplify $\frac{42!}{(32)! 10!}$ .

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Step 3.1

Rewrite $42!$ as $42 \cdot 41 \cdot 40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33 \cdot 32!$ .

$\frac{42 \cdot 41 \cdot 40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33 \cdot 32!}{(32)! 10!}$

Step 3.2

Cancel the common factor of $32!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{42 \cdot 41 \cdot 40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33 \cdot 32!}{(32)! 10!}$

Step 3.2.2

Rewrite the expression.

$\frac{42 \cdot 41 \cdot 40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $42$ by $41$ .

$\frac{1722 \cdot 40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.2

Multiply $1722$ by $40$ .

$\frac{68880 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.3

Multiply $68880$ by $39$ .

$\frac{2686320 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.4

Multiply $2686320$ by $38$ .

$\frac{102080160 \cdot 37 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.5

Multiply $102080160$ by $37$ .

$\frac{3776965920 \cdot 36 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.6

Multiply $3776965920$ by $36$ .

$\frac{135970773120 \cdot 35 \cdot 34 \cdot 33}{10!}$

Step 3.3.7

Multiply $135970773120$ by $35$ .

$\frac{4758977059200 \cdot 34 \cdot 33}{10!}$

Step 3.3.8

Multiply $4758977059200$ by $34$ .

$\frac{161805220012800 \cdot 33}{10!}$

Step 3.3.9

Multiply $161805220012800$ by $33$ .

$\frac{5339572260422400}{10!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand $10!$ to $10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{5339572260422400}{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply $10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply $10$ by $9$ .

$\frac{5339572260422400}{90 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply $90$ by $8$ .

$\frac{5339572260422400}{720 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply $720$ by $7$ .

$\frac{5339572260422400}{5040 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply $5040$ by $6$ .

$\frac{5339572260422400}{30240 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply $30240$ by $5$ .

$\frac{5339572260422400}{151200 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.6

Multiply $151200$ by $4$ .

$\frac{5339572260422400}{604800 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.7

Multiply $604800$ by $3$ .

$\frac{5339572260422400}{1814400 \cdot 2 \cdot 1}$

Step 3.4.2.8

Multiply $1814400$ by $2$ .

$\frac{5339572260422400}{3628800 \cdot 1}$

Step 3.4.2.9

Multiply $3628800$ by $1$ .

$\frac{5339572260422400}{3628800}$

Step 3.5

Divide $5339572260422400$ by $3628800$ .

$1471442973$