Relations
PYP_JEE_ADV_2024_P2
Grade None
Question:
PARAGRAPH I
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 $(a,b) \in R$, we have $|a-b| \ge 2$.
Let $Y = \{R \in X : \text{the range of } R \text{ has exactly one element}\}$ and $Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}$.
Let $n(A)$ denote the number of elements in a set $A$.
If $n(X) = {}^m C_6$, then the value of $m$ is ___.
Step-by-Step Solution
Key Concept: Counting valid elements in a relation matrix using complementary counting (subtracting diagonal and adjacent entries).
1) A relation $R$ from $S$ to $S$ is a subset of $S \times S$.
We need to find the number of pairs $(a, b) \in S \times S$ that satisfy $|a - b| \ge 2$.
Let's count the number of invalid pairs, which are those with $|a - b| < 2 \implies |a - b| = 0$ or $|a - b| = 1$:
- For $|a - b| = 0$: 6 pairs (diagonal elements).
- For $|a - b| = 1$: 10 pairs (adjacent elements: $(1,2), (2,1), (2,3), (3,2), \dots$).
Total invalid pairs = $6 + 10 = 16$.
Since $S \times S$ has $6 \times 6 = 36$ pairs, the number of valid pairs is:
$$36 - 16 = 20$$
2) Since $R \in X$ has exactly 6 elements chosen from these 20 valid pairs, the number of such relations is:
$$n(X) = \binom{20}{6} = {}^{20}C_6$$
Comparing this to ${}^mC_6$, we get:
$$m = 20$$
Thus, the answer is 20.
Correct Answer: 20