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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)5
- b)6
- c)7
- d)8
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The total number of subsets of a finite set A has 56 more elements, th...
Let sets A and B have m and n elements, respectively,
Then 2m - 2n = 56 (According to question)
implies2n(2m-n - 1)
= 8 x 7 = (2)3 x 7 = 23(23- l )
On comparing implies n = 3
and m-n = 3
implies m = 6 and n = 3
Number of subsets of A = 23 + 56 = 64 = 26
Hence,Number of elements of in A = 6
Then 2m - 2n = 56 (According to question)
implies2n(2m-n - 1)
= 8 x 7 = (2)3 x 7 = 23(23- l )
On comparing implies n = 3
and m-n = 3
implies m = 6 and n = 3
Number of subsets of A = 23 + 56 = 64 = 26
Hence,Number of elements of in A = 6
Most Upvoted Answer
The total number of subsets of a finite set A has 56 more elements, th...
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'.
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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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?
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