Calculus Examples

$({}_{14}C_{7} \cdot 8) ({}_{14}C_{7} \cdot 8)$

Step 1

Evaluate ${}_{14}C_{7}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$\frac{14!}{(14 - 7)! 7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 2

Subtract $7$ from $14$ .

$\frac{14!}{(7)! 7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3

Simplify $\frac{14!}{(7)! 7!}$ .

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Step 3.1

Rewrite $14!$ as $14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!$ .

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!}{(7)! 7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.2

Cancel the common factor of $7!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!}{(7)! 7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.2.2

Rewrite the expression.

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $14$ by $13$ .

$\frac{182 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3.2

Multiply $182$ by $12$ .

$\frac{2184 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3.3

Multiply $2184$ by $11$ .

$\frac{24024 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3.4

Multiply $24024$ by $10$ .

$\frac{240240 \cdot 9 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3.5

Multiply $240240$ by $9$ .

$\frac{2162160 \cdot 8}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.3.6

Multiply $2162160$ by $8$ .

$\frac{17297280}{7!} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4

Simplify the denominator.

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Step 3.4.1

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

$\frac{17297280}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

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 $6$ .

$\frac{17297280}{42 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4.2.2

Multiply $42$ by $5$ .

$\frac{17297280}{210 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4.2.3

Multiply $210$ by $4$ .

$\frac{17297280}{840 \cdot 3 \cdot 2 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4.2.4

Multiply $840$ by $3$ .

$\frac{17297280}{2520 \cdot 2 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4.2.5

Multiply $2520$ by $2$ .

$\frac{17297280}{5040 \cdot 1} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.4.2.6

Multiply $5040$ by $1$ .

$\frac{17297280}{5040} \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 3.5

Divide $17297280$ by $5040$ .

$3432 \cdot 8 ({}_{14}C_{7} \cdot 8)$

Step 4

Evaluate ${}_{14}C_{7}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$3432 \cdot 8 (\frac{14!}{(14 - 7)! 7!} \cdot 8)$

Step 5

Subtract $7$ from $14$ .

$3432 \cdot 8 (\frac{14!}{(7)! 7!} \cdot 8)$

Step 6

Simplify $\frac{14!}{(7)! 7!}$ .

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Step 6.1

Rewrite $14!$ as $14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!$ .

$3432 \cdot 8 (\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!}{(7)! 7!} \cdot 8)$

Step 6.2

Cancel the common factor of $7!$ .

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Step 6.2.1

Cancel the common factor.

$3432 \cdot 8 (\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7!}{(7)! 7!} \cdot 8)$

Step 6.2.2

Rewrite the expression.

$3432 \cdot 8 (\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8)$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $14$ by $13$ .

$3432 \cdot 8 (\frac{182 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8)$

Step 6.3.2

Multiply $182$ by $12$ .

$3432 \cdot 8 (\frac{2184 \cdot 11 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8)$

Step 6.3.3

Multiply $2184$ by $11$ .

$3432 \cdot 8 (\frac{24024 \cdot 10 \cdot 9 \cdot 8}{7!} \cdot 8)$

Step 6.3.4

Multiply $24024$ by $10$ .

$3432 \cdot 8 (\frac{240240 \cdot 9 \cdot 8}{7!} \cdot 8)$

Step 6.3.5

Multiply $240240$ by $9$ .

$3432 \cdot 8 (\frac{2162160 \cdot 8}{7!} \cdot 8)$

Step 6.3.6

Multiply $2162160$ by $8$ .

$3432 \cdot 8 (\frac{17297280}{7!} \cdot 8)$

Step 6.4

Simplify the denominator.

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Step 6.4.1

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

$3432 \cdot 8 (\frac{17297280}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8)$

Step 6.4.2

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

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Step 6.4.2.1

Multiply $7$ by $6$ .

$3432 \cdot 8 (\frac{17297280}{42 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8)$

Step 6.4.2.2

Multiply $42$ by $5$ .

$3432 \cdot 8 (\frac{17297280}{210 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 8)$

Step 6.4.2.3

Multiply $210$ by $4$ .

$3432 \cdot 8 (\frac{17297280}{840 \cdot 3 \cdot 2 \cdot 1} \cdot 8)$

Step 6.4.2.4

Multiply $840$ by $3$ .

$3432 \cdot 8 (\frac{17297280}{2520 \cdot 2 \cdot 1} \cdot 8)$

Step 6.4.2.5

Multiply $2520$ by $2$ .

$3432 \cdot 8 (\frac{17297280}{5040 \cdot 1} \cdot 8)$

Step 6.4.2.6

Multiply $5040$ by $1$ .

$3432 \cdot 8 (\frac{17297280}{5040} \cdot 8)$

Step 6.5

Divide $17297280$ by $5040$ .

$3432 \cdot 8 (3432 \cdot 8)$

Step 7

Multiply $3432 \cdot 8 (3432 \cdot 8)$ .

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Step 7.1

Multiply $3432$ by $8$ .

$27456 (3432 \cdot 8)$

Step 7.2

Multiply $3432$ by $8$ .

$27456 \cdot 27456$

Step 7.3

Multiply $27456$ by $27456$ .

$753831936$