If n items are arranged two at a time, the number obtained is 20. Find the value of n.
Question 1 Report
If n items are arranged two at a time, the number obtained is 20. Find the value of n.
Answer Details
The number of ways to arrange n items taken two at a time is given by the formula:
\(_nP_2 = \frac{n!}{(n-2)!} = n(n-1)\)
We are given that \(_nP_2 = 20\), so we can set up the equation:
\begin{aligned} n(n-1) &= 20 \\ n^2 - n - 20 &= 0 \end{aligned}
We can solve for n using the quadratic formula:
\begin{aligned} n &= \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)} \\ &= \frac{1 \pm \sqrt{81}}{2} \\ &= \frac{1 \pm 9}{2} \end{aligned}
We discard the negative solution, and the positive solution is:
\begin{aligned} n &= \frac{1+9}{2} \\ &= 5 \end{aligned}
Therefore, the value of n is 5.
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