Question

Two finite sets have m and n elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are

• A. 7, 6
• B. 5, 1
• C. 6, 3
• D. 8, 7

Answer 1

The Correct Option is A

Solution:

Step 1: Understanding the number of subsets:

The number of subsets of a set with k elements is given by the formula:
Number of subsets = 2^k

Step 2: Relating the given information:

Let the number of subsets of the first set with m elements be 2^m, and the number of subsets of the second set with n elements be 2ⁿ. According to the problem, the total number of subsets of the first set is 56 more than the total number of subsets of the second set:
2^m = 2ⁿ + 56

Step 3: Substituting and solving the equation:

We need to solve the equation 2^m = 2ⁿ + 56 to find the values of m and n. Try different values of m and n to satisfy this equation:

If we try m = 7 and n = 6, we get:
2⁷ = 128 and 2⁶ + 56 = 64 + 56 = 120
Since 128 = 120 + 56, this solution satisfies the equation.

Step 4: Conclusion:

Therefore, the values of m and n are 7 and 6 respectively.

Answer: The values of m and n are 7 and 6 respectively, which corresponds to Option 1.