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Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is : [JEE M 2015]
- a)275
- b)510
- c)219
- d)256
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let A and B be two sets containing four and two elements respectively....
n(A) = 4, n (B) = 2 ⇒ n(A × B) = 8
The number of subsets of A × B having atleast 3 = elements = 8C3 + 8C4 + ... + 8C8
= 28 – 8C0 – 8C1 – 8C2
= 256 – 1 – 8 – 28 = 219
The number of subsets of A × B having atleast 3 = elements = 8C3 + 8C4 + ... + 8C8
= 28 – 8C0 – 8C1 – 8C2
= 256 – 1 – 8 – 28 = 219
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Community Answer
Let A and B be two sets containing four and two elements respectively....
Understanding the Problem
To find the number of subsets of the set \( A \times B \) having at least three elements, we start by identifying the sizes of sets \( A \) and \( B \):
- Set \( A \) has 4 elements.
- Set \( B \) has 2 elements.
The Cartesian product \( A \times B \) will have \( |A| \times |B| = 4 \times 2 = 8 \) elements.
Counting the Total Subsets
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Thus, the total number of subsets of \( A \times B \) is:
- \( 2^8 = 256 \)
Counting Subsets with Fewer than 3 Elements
Next, we need to count and exclude subsets that have fewer than 3 elements:
- 0 Elements: There is 1 subset (the empty set).
- 1 Element: There are \( \binom{8}{1} = 8 \) subsets.
- 2 Elements: There are \( \binom{8}{2} = 28 \) subsets.
Now, we sum these subsets:
- Total subsets with fewer than 3 elements = \( 1 + 8 + 28 = 37 \)
Calculating Subsets with At Least 3 Elements
Now, we subtract the number of subsets with fewer than 3 elements from the total number of subsets:
- Subsets with at least 3 elements = Total subsets - Subsets with fewer than 3 elements
- \( 256 - 37 = 219 \)
Conclusion
Thus, the number of subsets of the set \( A \times B \) with at least three elements is:
- 219, which corresponds to option C.
To find the number of subsets of the set \( A \times B \) having at least three elements, we start by identifying the sizes of sets \( A \) and \( B \):
- Set \( A \) has 4 elements.
- Set \( B \) has 2 elements.
The Cartesian product \( A \times B \) will have \( |A| \times |B| = 4 \times 2 = 8 \) elements.
Counting the Total Subsets
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Thus, the total number of subsets of \( A \times B \) is:
- \( 2^8 = 256 \)
Counting Subsets with Fewer than 3 Elements
Next, we need to count and exclude subsets that have fewer than 3 elements:
- 0 Elements: There is 1 subset (the empty set).
- 1 Element: There are \( \binom{8}{1} = 8 \) subsets.
- 2 Elements: There are \( \binom{8}{2} = 28 \) subsets.
Now, we sum these subsets:
- Total subsets with fewer than 3 elements = \( 1 + 8 + 28 = 37 \)
Calculating Subsets with At Least 3 Elements
Now, we subtract the number of subsets with fewer than 3 elements from the total number of subsets:
- Subsets with at least 3 elements = Total subsets - Subsets with fewer than 3 elements
- \( 256 - 37 = 219 \)
Conclusion
Thus, the number of subsets of the set \( A \times B \) with at least three elements is:
- 219, which corresponds to option C.
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Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is : [JEE M 2015]a)275b)510c)219d)256Correct answer is option 'C'. Can you explain this answer?
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