Calculus Examples

${}_{5}P_{3} \cdot {}_{6}C_{4}$

Step 1

Evaluate ${}_{5}P_{3}$ using the formula ${}_{n}P_{r} = \frac{n!}{(n - r)!}$ .

$\frac{5!}{(5 - 3)!} \cdot {}_{6}C_{4}$

Step 2

Subtract $3$ from $5$ .

$\frac{5!}{(2)!} \cdot {}_{6}C_{4}$

Step 3

Simplify $\frac{5!}{(2)!}$ .

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Step 3.1

Rewrite $5!$ as $5 \cdot 4 \cdot 3 \cdot 2!$ .

$\frac{5 \cdot 4 \cdot 3 \cdot 2!}{(2)!} \cdot {}_{6}C_{4}$

Step 3.2

Cancel the common factor of $2!$ .

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Step 3.2.1

Cancel the common factor.

$\frac{5 \cdot 4 \cdot 3 \cdot 2!}{(2)!} \cdot {}_{6}C_{4}$

Step 3.2.2

Divide $5 \cdot 4 \cdot 3$ by $1$ .

$5 \cdot 4 \cdot 3 \cdot {}_{6}C_{4}$

Step 3.3

Multiply $5 \cdot 4 \cdot 3$ .

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Step 3.3.1

Multiply $5$ by $4$ .

$20 \cdot 3 \cdot {}_{6}C_{4}$

Step 3.3.2

Multiply $20$ by $3$ .

$60 \cdot {}_{6}C_{4}$

Step 4

Evaluate ${}_{6}C_{4}$ using the formula ${}_{n}C_{r} = \frac{n!}{(n - r)! r!}$ .

$60 \cdot \frac{6!}{(6 - 4)! 4!}$

Step 5

Subtract $4$ from $6$ .

$60 \cdot \frac{6!}{(2)! 4!}$

Step 6

Simplify $\frac{6!}{(2)! 4!}$ .

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Step 6.1

Rewrite $6!$ as $6 \cdot 5 \cdot 4!$ .

$60 \cdot \frac{6 \cdot 5 \cdot 4!}{(2)! 4!}$

Step 6.2

Cancel the common factor of $4!$ .

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Step 6.2.1

Cancel the common factor.

$60 \cdot \frac{6 \cdot 5 \cdot 4!}{(2)! 4!}$

Step 6.2.2

Rewrite the expression.

$60 \cdot \frac{6 \cdot 5}{(2)!}$

Step 6.3

Multiply $6$ by $5$ .

$60 \cdot \frac{30}{(2)!}$

Step 6.4

Simplify the denominator.

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Step 6.4.1

Expand $(2)!$ to $2 \cdot 1$ .

$60 \cdot \frac{30}{2 \cdot 1}$

Step 6.4.2

Multiply $2$ by $1$ .

$60 \cdot \frac{30}{2}$

Step 6.5

Divide $30$ by $2$ .

$60 \cdot 15$

Step 7

Multiply $60$ by $15$ .

$900$