Finite Math Examples

${}_{990}C_{10}$

Step 1

Evaluate ${}_{990}C_{10}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{990!}{(990 - 10)! 10!}$

Step 2

Subtract $10$ from $990$ .

$\frac{990!}{(980)! 10!}$

Step 3

Simplify $\frac{990!}{(980)! 10!}$ .

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Step 3.1

Rewrite $990!$ as $990 \cdot 989 \cdot 988 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981 \cdot 980!$ .

$\frac{990 \cdot 989 \cdot 988 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981 \cdot 980!}{(980)! 10!}$

Step 3.2

Cancel the common factor of $980!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{990 \cdot 989 \cdot 988 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981 \cdot 980!}{(980)! 10!}$

Step 3.2.2

Rewrite the expression.

$\frac{990 \cdot 989 \cdot 988 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $990$ by $989$ .

$\frac{979110 \cdot 988 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.2

Multiply $979110$ by $988$ .

$\frac{967360680 \cdot 987 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.3

Multiply $967360680$ by $987$ .

$\frac{954784991160 \cdot 986 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.4

Multiply $954784991160$ by $986$ .

$\frac{941418001283760 \cdot 985 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.5

Multiply $941418001283760$ by $985$ .

$\frac{927296731264503600 \cdot 984 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.6

Multiply $927296731264503600$ by $984$ .

$\frac{912459983564271542400 \cdot 983 \cdot 982 \cdot 981}{10!}$

Step 3.3.7

Multiply $912459983564271542400$ by $983$ .

$\frac{896948163843678926179200 \cdot 982 \cdot 981}{10!}$

Step 3.3.8

Multiply $896948163843678926179200$ by $982$ .

$\frac{880803096894492705507974400 \cdot 981}{10!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

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

$\frac{880803096894492705507974400 \cdot 981}{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

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 $9$ .

$\frac{880803096894492705507974400 \cdot 981}{90 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply $90$ by $8$ .

$\frac{880803096894492705507974400 \cdot 981}{720 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply $720$ by $7$ .

$\frac{880803096894492705507974400 \cdot 981}{5040 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply $5040$ by $6$ .

$\frac{880803096894492705507974400 \cdot 981}{30240 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply $30240$ by $5$ .

$\frac{880803096894492705507974400 \cdot 981}{151200 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.6

Multiply $151200$ by $4$ .

$\frac{880803096894492705507974400 \cdot 981}{604800 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.7

Multiply $604800$ by $3$ .

$\frac{880803096894492705507974400 \cdot 981}{1814400 \cdot 2 \cdot 1}$

Step 3.4.2.8

Multiply $1814400$ by $2$ .

$\frac{880803096894492705507974400 \cdot 981}{3628800 \cdot 1}$

Step 3.4.2.9

Multiply $3628800$ by $1$ .

$\frac{880803096894492705507974400 \cdot 981}{3628800}$

Step 3.5

Cancel the common factor of $880803096894492705507974400$ and $3628800$ .

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Step 3.5.1

Factor $3628800$ out of $880803096894492705507974400 \cdot 981$ .

$\frac{3628800 (242725721145969109763 \cdot 981)}{3628800}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $3628800$ out of $3628800$ .

$\frac{3628800 (242725721145969109763 \cdot 981)}{3628800 (1)}$

Step 3.5.2.2

Cancel the common factor.

$\frac{3628800 (242725721145969109763 \cdot 981)}{3628800 \cdot 1}$

Step 3.5.2.3

Rewrite the expression.

$\frac{242725721145969109763 \cdot 981}{1}$

Step 3.5.2.4

Divide $242725721145969109763 \cdot 981$ by $1$ .

$242725721145969109763 \cdot 981$

Step 3.6

Multiply $242725721145969109763$ by $981$ .

$238113932444195696677503$