Question

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

Hint:

For functions, consider all mappings that satisfy the given conditions and compute the total possible arrangements.

Answer 1

To compute \( n(Y) \): \[ n(Y) = 0 \quad \text{(Since the range of \( R \) has exactly one element, \( R \) cannot have 6 elements).} \] To compute \( n(Z) \): \[ n(Z) = \text{Number of functions from \( S \) to \( S \)} = \binom{4}{1} \cdot \binom{3}{1} \cdot \binom{3}{1} \cdot \binom{3}{1} \cdot \binom{3}{1} \cdot \binom{4}{1}. \] Calculating: \[ n(Z) = 36^2. \] Thus: \[ n(Y) + n(Z) = 36^2 \quad \Rightarrow \quad |k| = 36. \]