Comprehension
Let S = {1, 2, 3, 4, 5, 6} and X be the set of all relations R from S to S that satisfy both the following properties :
i. R has exactly 6 elements.
ii. For each (a, b) ∈ R, we have |a - b| ≥ 2.
Let Y = {R ∈ X : The range of R has exactly one element} and
Z = {R ∈ X : R is a function from S to S}.
Let n(A) denote the number of elements in a set A.
i. R has exactly 6 elements.
ii. For each (a, b) ∈ R, we have |a - b| ≥ 2.
Let Y = {R ∈ X : The range of R has exactly one element} and
Z = {R ∈ X : R is a function from S to S}.
Let n(A) denote the number of elements in a set A.
Question: 1
If n(X) = mC6, then the value of m is _______.
If n(X) = mC6, then the value of m is _______.
Updated On: Mar 8, 2025
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Correct Answer: 20
Solution and Explanation
Condition Satisfaction
To satisfy the condition \( |a - b| \geq 2 \), \( b \) must be at least 2 units away from \( a \). The possible pairs are:
- a = 1, b = 3, 4, 5, 6
- a = 2, b = 4, 5, 6
- a = 3, b = 1, 5, 6
- a = 4, b = 1, 2, 6
- a = 5, b = 1, 2, 3
- a = 6, b = 1, 2, 3, 4
Counting all valid pairs:
\[ \text{Total pairs} = 20 \]
Quick Tip:
- \( n(X) = 20 \)
- \( m = 20 \)
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Question: 2
If the value of n(Y) + n(Z) is k2, then |k| is _______.
If the value of n(Y) + n(Z) is k2, then |k| is _______.
Updated On: Mar 8, 2025
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Correct Answer: 36
Solution and Explanation
Calculating \( n(Y) \) and \( n(Z) \)
To calculate \( n(Y) \): Since the range of \( R \) has exactly one element, \( R \) cannot have 6 elements. Therefore:
\[ n(Y) = 0 \]
To calculate \( n(Z) \): The number of functions from \( S \) to \( S \) is determined by the product of choices for each mapping. Using the binomial coefficients:
\[ n(Z) = 4 \times 3 \times 3 \times 3 \times 3 \times 4 \]
Simplify:
\[ n(Z) = 362 \]
Thus, the total is:
\[ n(Y) + n(Z) = 362 \]
From this, we find:
\[ |k| = 36 \]
Final Answer:
- \( |k| = 36 \)
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