A quality control inspector randomly selects 7 distinct items from a batch of 20 items that are sequentially numbered from 1 to 20 . The probability that the median of the serial numbers of the selected items is exactly 12 is

Step-by-step solution

Let the selected serial numbers be arranged in increasing order as x₁ Since 7 items are selected, the median is the 4^ th item in the sorted sequence, which is x₄ . We are given that x₄ = 12 . For this to happen, the first 3 items ( x₁, x₂, x₃ ) must be selected from the numbers strictly less than 12 , which are the integers from 1 to 11 . The number of ways to select these 3 items is ¹¹C₃ . The last 3 items ( x₅, x₆, x₇ ) must be selected from the numbers strictly greater than 12 , which are the integers from 13 t