Question

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 \( m-n \)

• A. 4
• B. 6
• C. 8
• D. 1

Hint:

For a set with \( n \) elements, total subsets are \( 2^n \) and proper subsets are \( 2^n-1 \)Use these formulas directly in set-counting problems.

Answer 1

The Correct Option is A

Step 1: Recall number of subsets.
If a set has \( m \) elements, then total number of subsets is:
\[ 2^m \]
Number of proper subsets is:
\[ 2^m - 1 \]

Step 2: Write number of subsets of second set.
If second set has \( n \) elements, then total number of subsets is:
\[ 2^n \]

Step 3: Use the given condition.
Proper subsets of first set are 119 more than subsets of second set.
So:
\[ 2^m - 1 = 2^n + 119 \]

Step 4: Simplify the equation.
\[ 2^m - 2^n = 120 \]

Step 5: Factor the left-hand side.
\[ 2^n(2^{m-n} - 1) = 120 \]

Step 6: Express 120 suitably.
\[ 120 = 8 \times 15 \]
Also:
\[ 8 = 2^3 \]
and
\[ 15 = 2^4 - 1 \]
So:
\[ 2^n = 8 \]
\[ n = 3 \]
and
\[ 2^{m-n} - 1 = 15 \]
\[ 2^{m-n} = 16 \]
\[ m-n = 4 \]

Step 7: Final conclusion.
\[ \boxed{4} \]