JEE Advanced 2024 Paper 2 · Q14 · Permutations & Combinations
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 ______.
Reveal answer + step-by-step solution
Correct answer:20.00
Solution
We count pairs $(a, b) \in S \times S$ with $|a - b| \ge 2$. For each $a$, count $b \in S$ with $|a-b| \ge 2$: $a=1 \to b \in \{3,4,5,6\}$ (4); $a=2 \to \{4,5,6\}$ (3); $a=3 \to \{1,5,6\}$ (3); $a=4 \to \{1,2,6\}$ (3); $a=5 \to \{1,2,3\}$ (3); $a=6 \to \{1,2,3,4\}$ (4). Total $= 4+3+3+3+3+4 = 20$. Each element of $X$ is a $6$-element subset of these $20$ pairs, so $n(X) = \binom{20}{6} = {}^{20}C_{6}$. Hence $m = 20$.
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