If n ( X ) = C 6 n , then the value of m is [2024]

Question

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 $| a - b |$ $\geq 2$ .

Let $Y = {R \in X : The range of R has exactly one element}$ and

$Z = {R \in X : R is a function from S to S} .$

Let $n (A)$ denote the number of elements in a set A. [2024]

Q. If $n (X) = {}^{n}C_{6}$ , then the value of $m$ is _______.

Smart Achievers · Accepted Answer

(20)

Given $S = {1, 2, 3, 4, 5, 6}$

Let $R : S \to S$ be a relation such that

Number of elements in $R = 6$

and for each $(a, b) \in R :$ $| a - b |$ $\geq 2$

$X \to$ set of all such relations $R : S \to S$

Total number of ordered pairs $Z = {R \in X : R is a function from S to S} .$ such that $| a - b |$ $\geq 2 = 20$

$∴$ $n (X)$ = number of elements in $n (X) = {}^{n}C_{6}$

$∴$ $m = 20$

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