Two finite sets have `m` and `3` elements respectively. The total number of subsets of first set is `56` more than the total number of subsets of the second set. The value of `m` is ___________

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We have two finite sets, one with `m` elements (let's call it Set A) and another with `3` elements (let's call it Set B). We need to find the value of `m` given that the total number of subsets of Set A is `56` more than the total number of subsets of Set B.

### Step 2: Determine the formula for subsets
The total number of subsets of a set with `n` elements is given by the formula \(2^n\).

### Step 3: Calculate the number of subsets for both sets
- For Set A (with `m` elements), the number of subsets is \(2^m\).
- For Set B (with `3` elements), the number of subsets is \(2^3 = 8\).

### Step 4: Set up the equation based on the problem statement
According to the problem, the number of subsets of Set A is `56` more than the number of subsets of Set B. This can be expressed as:
\[ 2^m = 2^3 + 56 \]

### Step 5: Substitute the values
Substituting the value of \(2^3\):
\[ 2^m = 8 + 56 \]
\[ 2^m = 64 \]

### Step 6: Express 64 as a power of 2
We know that:
\[ 64 = 2^6 \]
So we can rewrite the equation as:
\[ 2^m = 2^6 \]

### Step 7: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ m = 6 \]

### Conclusion
The value of \(m\) is \(6\).

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