If the value of n(Y)+n(Z) is k^2 , then |k| is 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 ).

Step-by-step solution

given |a-b| 2 so if i.e. Total elements in X is ²⁰ C ₆ Now for n ( Y ) , range of R has exactly one element i.e. second elements must be constant in R and since R must have 6 element so it is not possible to satisfy both condition so n ( Y )=0 . aligned & for n(z) & 1 3,4,5,6 & 2 4,5,6 & 3 1,5,6 & 4 1,2,6 & 5 1,2,3 & 6 1,2,3,4 & aligned no. of relation that are function will be = ^4 C ₁ ^3 C ₁ ^3 C ₁ ^3 C ₁ ^3 C ₁ ^4 C ₁ aligned & =(4 3 3)^2=k^2 & i.e. k =36 aligned