A coffee roaster has 12 rare coffee beans with intensity scores ranked from 1 (mildest) to 12 (strongest) . You choose 7 beans at random and line them up from mildest to strongest: C₁ What is the probability that the third bean ( C₃ ) has an intensity score of exactly 4 ?

Step-by-step solution

The total number of ways to select 7 beans from 12 is ¹²C₇ . For the third bean C₃ to have an intensity score of exactly 4, the selection must satisfy the following conditions: Two beans must be chosen from the 3 beans with scores less than 4 (scores 1, 2, 3). The number of ways is ³C₂ . The bean with score 4 must be chosen. The number of ways is 1. Four beans must be chosen from the 8 beans with scores greater than 4 (scores 5 to 12). The number of ways is ⁸C₄ . The number of favorable selections is ³C₂ 1 ⁸C₄ . Th