The total number of subsets of a finite set A has 56 more elements, then the total number of subsets of another finite set B. What is the number of elements in the set A?a)5b)6c)7d)8Correct answer is option 'B'. Can you explain this answer?

Question

Answer

To solve this problem, we need to determine the number of elements in set A that satisfies the given condition.

Let's assume the number of elements in set A is 'n'.

The total number of subsets of a set with 'n' elements can be calculated using the formula 2^n. Similarly, the total number of subsets of set B would be 2^(n-3) since it has 3 fewer elements than set A.

According to the given condition, the total number of subsets of set A is 56 more than the total number of subsets of set B.

So, we can set up the equation as follows:

2^n = 2^(n-3) + 56

To simplify the equation, we can rewrite it as:

2^n = 2^n * 2^(-3) + 56

Since 2^(-3) is equal to 1/2^3 = 1/8, we can substitute it back into the equation:

2^n = 2^n/8 + 56

Next, we can multiply both sides of the equation by 8 to eliminate the fraction:

8 * 2^n = 2^n + 448

Now, we can subtract 2^n from both sides of the equation:

7 * 2^n = 448

Dividing both sides of the equation by 7:

2^n = 64

Now, we can determine the value of 'n' by finding the exponent that gives us 64.

We know that 2^6 = 64, so 'n' must be 6.

Therefore, the number of elements in set A is 6, which corresponds to option 'B'.

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