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.