Two finite sets have ' m ' and ' n ' number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of m and n are respectively.
To solve this problem, let's start by analyzing the given information. If a set has m elements, the total number of subsets of that set is 2m. Similarly, if another set has n elements, the total number of subsets of that set is 2n. According to the problem, the total number of subsets of the first set is 112 more than the total number of subsets of the second set. Therefore, we can write the equation: 2m=2n+112 We need to find the values of m and n that satisfy this equation. Let's check the given options one by one: Option A: m=7,n=4 Substitute these values into the equation: 27=24+112 Calculate the powers of 2: 128=16+112 This simplifies to: 128=128 This is a true statement, so Option A is correct. Option B: m=7,n=7 Substitute these values into the equation: 27=27+112 Calculate the powers of 2: 128=128+112 This simplifies to: 128=240 This is not true, so Option B is incorrect. Option C: m=4,n=4 Substitute these values into the equation: 24=24+112 Calculate the powers of 2 : 16=16+112 This simplifies to: 16=128 This is not true, so Option C is incorrect. Option D: m=4,n=7 Substitute these values into the equation: 24=27+112 Calculate the powers of 2: 16=128+112 This simplifies to: 16=240 This is not true, so Option D is incorrect. Therefore, the correct answer is Option A: m=7,n=4.