A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let X be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If is the mean of the random variable X , then the value of 77 is ___________.

Step-by-step solution

Total number of ways to choose 6 books from 11 books is ¹¹C₆ = 462 . Let M and P be the number of Mathematics and Physics books chosen respectively. Since 6 books are chosen, M + P = 6 . The random variable X is given by X = |M - P| = |2M - 6| . The possible selections of (M, P) , the corresponding values of X , and the number of ways are as follows: For M=1, P=5 : X = |1 - 5| = 4 , Number of ways = ⁶C₁ ⁵C₅ = 6 1 = 6 For M=2, P=4 : X = |2 - 4| = 2 , Number of ways = ⁶C₂ ⁵C₄ = 15 5 = 75 For M=3, P=3 : X = |3 - 3| =