Precalculus Examples

$1 - \frac{{}_{6}P_{4}}{{}_{8}P_{6}}$

Step 1

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

$1 - \frac{\frac{6!}{(6 - 4)!}}{{}_{8}P_{6}}$

Step 2

Subtract $4$ from $6$ .

$1 - \frac{\frac{6!}{(2)!}}{{}_{8}P_{6}}$

Step 3

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

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Step 3.1

Rewrite $6!$ as $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!$ .

$1 - \frac{\frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!}{(2)!}}{{}_{8}P_{6}}$

Step 3.2

Cancel the common factor of $2!$ .

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Step 3.2.1

Cancel the common factor.

$1 - \frac{\frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!}{(2)!}}{{}_{8}P_{6}}$

Step 3.2.2

Divide $6 \cdot 5 \cdot 4 \cdot 3$ by $1$ .

$1 - \frac{6 \cdot 5 \cdot 4 \cdot 3}{{}_{8}P_{6}}$

Step 3.3

Multiply $6 \cdot 5 \cdot 4 \cdot 3$ .

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Step 3.3.1

Multiply $6$ by $5$ .

$1 - \frac{30 \cdot 4 \cdot 3}{{}_{8}P_{6}}$

Step 3.3.2

Multiply $30$ by $4$ .

$1 - \frac{120 \cdot 3}{{}_{8}P_{6}}$

Step 3.3.3

Multiply $120$ by $3$ .

$1 - \frac{360}{{}_{8}P_{6}}$

Step 4

Evaluate ${}_{8}P_{6}$ using the formula ${}_{n}P_{r} = \frac{n!}{(n - r)!}$ .

$1 - \frac{360}{\frac{8!}{(8 - 6)!}}$

Step 5

Subtract $6$ from $8$ .

$1 - \frac{360}{\frac{8!}{(2)!}}$

Step 6

Simplify $\frac{8!}{(2)!}$ .

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Step 6.1

Rewrite $8!$ as $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!$ .

$1 - \frac{360}{\frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!}{(2)!}}$

Step 6.2

Cancel the common factor of $2!$ .

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Step 6.2.1

Cancel the common factor.

$1 - \frac{360}{\frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2!}{(2)!}}$

Step 6.2.2

Divide $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3$ by $1$ .

$1 - \frac{360}{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3}$

Step 6.3

Multiply $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3$ .

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Step 6.3.1

Multiply $8$ by $7$ .

$1 - \frac{360}{56 \cdot 6 \cdot 5 \cdot 4 \cdot 3}$

Step 6.3.2

Multiply $56$ by $6$ .

$1 - \frac{360}{336 \cdot 5 \cdot 4 \cdot 3}$

Step 6.3.3

Multiply $336$ by $5$ .

$1 - \frac{360}{1680 \cdot 4 \cdot 3}$

Step 6.3.4

Multiply $1680$ by $4$ .

$1 - \frac{360}{6720 \cdot 3}$

Step 6.3.5

Multiply $6720$ by $3$ .

$1 - \frac{360}{20160}$

Step 7

Cancel the common factor of $360$ and $20160$ .

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Step 7.1

Factor $360$ out of $360$ .

$1 - \frac{360 (1)}{20160}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $360$ out of $20160$ .

$1 - \frac{360 \cdot 1}{360 \cdot 56}$

Step 7.2.2

Cancel the common factor.

$1 - \frac{360 \cdot 1}{360 \cdot 56}$

Step 7.2.3

Rewrite the expression.

$1 - \frac{1}{56}$

Step 8

Simplify the expression.

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Step 8.1

Write $1$ as a fraction with a common denominator.

$\frac{56}{56} - \frac{1}{56}$

Step 8.2

Combine the numerators over the common denominator.

$\frac{56 - 1}{56}$

Step 8.3

Subtract $1$ from $56$ .

$\frac{55}{56}$