For the next three (03) items that follow : Five numbers are randomly picked from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and arranged them in ascending order [\(x_1 < x_2 < x_3 < x_4 < x_5\); \(x_1\) being the smallest number and \(x_5\) being the largest number selected].
What is \(P(x_3 = 6)\) equal to?
\(\dfrac{5}{21}\)
For \(x_3=6\), exactly 2 numbers must be chosen from \(\{1,2,3,4,5\}\), the number 6 itself, and exactly 2 numbers from \(\{7,8,9,10\}\). This gives \(\binom{5}{2}\binom{4}{2}=10\times 6=60\) favourable selections out of \(\binom{10}{5}=252\), so \(P(x_3=6) = \dfrac{60}{252} = \dfrac{5}{21}\).
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