Finite Math Examples

${}_{25}C_{11}$

Step 1

Evaluate ${}_{25}C_{11}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{25!}{(25 - 11)! 11!}$

Step 2

Subtract $11$ from $25$ .

$\frac{25!}{(14)! 11!}$

Step 3

Simplify $\frac{25!}{(14)! 11!}$ .

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Step 3.1

Rewrite $25!$ as $25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14!$ .

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14!}{(14)! 11!}$

Step 3.2

Cancel the common factor of $14!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14!}{(14)! 11!}$

Step 3.2.2

Rewrite the expression.

$\frac{25 \cdot 24 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $25$ by $24$ .

$\frac{600 \cdot 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.2

Multiply $600$ by $23$ .

$\frac{13800 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.3

Multiply $13800$ by $22$ .

$\frac{303600 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.4

Multiply $303600$ by $21$ .

$\frac{6375600 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.5

Multiply $6375600$ by $20$ .

$\frac{127512000 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.6

Multiply $127512000$ by $19$ .

$\frac{2422728000 \cdot 18 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.7

Multiply $2422728000$ by $18$ .

$\frac{43609104000 \cdot 17 \cdot 16 \cdot 15}{11!}$

Step 3.3.8

Multiply $43609104000$ by $17$ .

$\frac{741354768000 \cdot 16 \cdot 15}{11!}$

Step 3.3.9

Multiply $741354768000$ by $16$ .

$\frac{11861676288000 \cdot 15}{11!}$

Step 3.3.10

Multiply $11861676288000$ by $15$ .

$\frac{177925144320000}{11!}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Expand $11!$ to $11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{177925144320000}{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2

Multiply $11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.4.2.1

Multiply $11$ by $10$ .

$\frac{177925144320000}{110 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.2

Multiply $110$ by $9$ .

$\frac{177925144320000}{990 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.3

Multiply $990$ by $8$ .

$\frac{177925144320000}{7920 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.4

Multiply $7920$ by $7$ .

$\frac{177925144320000}{55440 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.5

Multiply $55440$ by $6$ .

$\frac{177925144320000}{332640 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.6

Multiply $332640$ by $5$ .

$\frac{177925144320000}{1663200 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.7

Multiply $1663200$ by $4$ .

$\frac{177925144320000}{6652800 \cdot 3 \cdot 2 \cdot 1}$

Step 3.4.2.8

Multiply $6652800$ by $3$ .

$\frac{177925144320000}{19958400 \cdot 2 \cdot 1}$

Step 3.4.2.9

Multiply $19958400$ by $2$ .

$\frac{177925144320000}{39916800 \cdot 1}$

Step 3.4.2.10

Multiply $39916800$ by $1$ .

$\frac{177925144320000}{39916800}$

Step 3.5

Divide $177925144320000$ by $39916800$ .

$4457400$