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A is a set containing n elements, subset P of A is chosen. The set A is reconstructed by replacing the elements of P, subset Q of A is chosen. The number of ways of selecting P and Q so that P and Q are non intersecting, is
  • a)
    2n − 1
  • b)
    3n − 1
  • c)
    3n − 2
  • d)
    3n + 1
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A is a set containing n elements, subset P of A is chosen. The set A ...
Problem:
Given a set A with n elements, we need to choose two subsets P and Q such that P and Q are non-intersecting.

Approach:
To solve this problem, we can use the concept of Inclusion-Exclusion Principle.

Inclusion-Exclusion Principle:
The Inclusion-Exclusion Principle is a counting technique used to calculate the size of a union of sets. It states that the size of the union of sets is equal to the sum of the sizes of individual sets, minus the sum of the sizes of all possible intersections of pairs of sets, plus the sum of the sizes of all possible intersections of triples of sets, and so on.

Step 1: Selecting the subset P
- We have n elements in set A.
- For each element in A, we have two choices - either include it in P or exclude it from P.
- So, the total number of ways to select P is 2^n.

Step 2: Selecting the subset Q
- After selecting P, the remaining elements in set A are (n - |P|), where |P| denotes the number of elements in P.
- For each element in the remaining set, we have three choices - include it in Q, exclude it from Q, or include it in P and Q (which is not allowed as per the problem).
- So, the total number of ways to select Q is 3^(n - |P|).

Step 3: Counting the number of non-intersecting subsets
- For each selection of P, we have 2^n choices for P and 3^(n - |P|) choices for Q.
- However, we need to ensure that P and Q are non-intersecting.
- To count the number of non-intersecting subsets, we need to subtract the cases where P and Q intersect.
- The number of ways P and Q can intersect is the number of ways to select a subset from P.
- Since P can have |P| elements, the number of ways to select a subset from P is 2^|P|.
- Therefore, the number of non-intersecting subsets is 2^n - 2^|P|.

Step 4: Simplifying the expression
- Since we need to find the total number of ways to select P and Q, we can sum the number of non-intersecting subsets over all possible values of |P|.
- The value of |P| can range from 0 to n.
- So, the total number of ways to select P and Q is given by the sum of (2^n - 2^|P|) for |P| from 0 to n.
- This can be simplified as follows:
- 2^n - 2^0 + 2^n - 2^1 + 2^n - 2^2 + ... + 2^n - 2^n
- = n * 2^n - (2^0 + 2^1 + 2^2 + ... + 2^n)
- = n * 2^n - (2^(n+1) - 1)
- = n * 2^n - 2^(n+1) + 1
-
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Community Answer
A is a set containing n elements, subset P of A is chosen. The set A ...
Let A = {a1, a2, a3......an}P, Q are subsets of A elements in P, Q can be chosen in following ways
Case I: ai ∈ P ai ∈ Q i = 1,2,...n
Case II: ai ∈ P ai ∉ Q i = 1,2,....n
Case III: ai ∉ P ai ∈ Q i = 1,2,....n
Case IV: ai ∉ P ai ∉ Q i = 1,2,...n
∵ P, Q are non intersecting sets every element can be chosen as in case (II), case (III), case (IV) i.e. every elements can be chosen in 3 ways. Total number of ways
= 3 × 3.....n = times = 3n ways
But one case needs to be excluded when P = ϕ, Q = ϕ
∴ Number of ways = 3n − 1
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A is a set containing n elements, subset P of A is chosen. The set A is reconstructed by replacing the elements of P, subset Q of A is chosen. The number of ways of selecting P and Q so that P and Q are non intersecting, isa)2n − 1b)3n − 1c)3n − 2d)3n + 1Correct answer is option 'B'. Can you explain this answer?
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A is a set containing n elements, subset P of A is chosen. The set A is reconstructed by replacing the elements of P, subset Q of A is chosen. The number of ways of selecting P and Q so that P and Q are non intersecting, isa)2n − 1b)3n − 1c)3n − 2d)3n + 1Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A is a set containing n elements, subset P of A is chosen. The set A is reconstructed by replacing the elements of P, subset Q of A is chosen. The number of ways of selecting P and Q so that P and Q are non intersecting, isa)2n − 1b)3n − 1c)3n − 2d)3n + 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A is a set containing n elements, subset P of A is chosen. The set A is reconstructed by replacing the elements of P, subset Q of A is chosen. The number of ways of selecting P and Q so that P and Q are non intersecting, isa)2n − 1b)3n − 1c)3n − 2d)3n + 1Correct answer is option 'B'. Can you explain this answer?.
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