Given that \(^{n}P_{r} = 90\) and \(^{n}C_{r} = 15\), find the value of r.
Question 1 Report
Given that \(^{n}P_{r} = 90\) and \(^{n}C_{r} = 15\), find the value of r.
Answer Details
We know that:
$$^{n}P_{r} = \frac{n!}{(n-r)!} = 90$$
and
$$^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!(n-r)!} = 15$$
To find the value of r, we can use the formula:
$$^{n}C_{r} = \frac{^{n}P_{r}}{r!}$$
Substituting the given values, we get:
$$15 = \frac{90}{r!}$$
Simplifying the equation, we get:
$$r! = 6$$
The only integer value of r that satisfies this equation is 3, since 3! = 6. Therefore, the answer is r = 3.
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