Question:
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.
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.
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To solve such problems, express the total number of subsets as powers of 2, then use the given difference to form an equation and solve for \( m \) and \( n \).
Updated On: Apr 16, 2025
- \( 7, 4 \)
- \( 7, 7 \)
- \( 4, 4 \)
- \( 4, 7 \)
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The Correct Option is A
Solution and Explanation
The total number of subsets of a set with \( n \) elements is given by \( 2^n \).
Let the number of elements in the first set be \( m \) and in the second set be \( n \).
Thus, the number of subsets of the first set is \( 2^m \), and the number of subsets of the second set is \( 2^n \).
According to the given condition, we have the equation:
\[
2^m = 2^n + 112
\]
Now, we need to find the values of \( m \) and \( n \) that satisfy this equation.
First, test for \( m = 7 \) and \( n = 4 \):
\[
2^7 = 128, \quad 2^4 = 16
\]
Substitute these into the equation:
\[
128 = 16 + 112
\]
This is true, so the values of \( m \) and \( n \) are \( 7 \) and \( 4 \), respectively.
Thus, the correct answer is \( (A) \).
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