If \(\frac{^{8}P_{x}}{^{8}C_{x}} = 6\), find the value of x.
Answer Details
The formula for permutations is:
\(^{n}P_{r} = \frac{n!}{(n-r)!}\)
The formula for combinations is:
\(^{n}C_{r} = \frac{n!}{r!(n-r)!}\)
We are given that:
\(\frac{^{8}P_{x}}{^{8}C_{x}} = 6\)
Substituting the formulas for permutations and combinations, we get:
\(\frac{\frac{8!}{(8-x)!}}{\frac{8!}{x!(8-x)!}} = 6\)
Simplifying, we get:
\(\frac{x!}{(8-x)!} = \frac{1}{6}\)
Multiplying both sides by \(\frac{(8-x)!}{x!}\), we get:
\(\frac{8!}{x!(8-x)!} = 6\)
Dividing both sides by 8!, we get:
\(\frac{1}{^{8}C_{x}} = \frac{1}{720}\)
Multiplying both sides by 720, we get:
\(^{8}C_{x} = 720\)
We can use the formula for combinations to find the value of x:
\(^{8}C_{x} = \frac{8!}{x!(8-x)!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{x!(8-x)!}\)
Since \(^{8}C_{x} = 720\), we have:
\(\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{x!(8-x)!} = 720\)
Dividing both sides by 720, we get:
\(\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{720} = x!(8-x)!\)
Simplifying, we get:
\(x!(8-x)! = 40320\)
The only value of x that satisfies this equation is x = 3. Therefore, the correct answer is.