Two finite sets have m and n elements. The total number of proper subsets of the first set is 119 more than the total number of subsets of the second set. Find the value of m-n

Step-by-step solution

The number of subsets of a set with m elements is 2^m . The number of proper subsets of the first set is 2^m - 1 . The number of subsets of the second set is 2^n . According to the problem, 2^m - 1 = 2^n + 119 . Rearranging the equation, we get 2^m - 2^n = 120 . Factoring out 2^n , we have 2^n(2^ m-n - 1) = 120 . Expressing 120 as a product of a power of 2 and an odd number, 120 = 8 15 = 2^3 (2^4 - 1) . Comparing the two sides, we get n = 3 and m - n = 4 . Therefore, m - n = 4 . Answer: 4