QJEEM12Mab73e: JEE Main 2012 Mathematics - Class 11 JEEM12Mab73e Solution: Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs…

For any element x_i present in X, 4 cases arises while making subsets Y and Z.

Case- 1 : x_i $$\in$$ Y, x_i $$\in$$ Z $$\Rightarrow$$ Y $$\cap$$ Z $$\ne$$ $$\varphi$$

Case- 2 : x_i $$\in$$ Y, x_i $$\notin$$ Z $$\Rightarrow$$ Y $$\cap$$ Z = $$\varphi$$

Case- 3 : x_i $$\notin$$ Y, x_i $$\in$$ Z $$\Rightarrow$$ Y $$\cap$$ Z = $$\varphi$$

Case- 4 : x_i $$\notin$$ Y, x_i $$\notin$$ Z $$\Rightarrow$$ Y $$\cap$$ Z = $$\varphi$$

$$\therefore$$ For every element, number of ways = 3 for which Y $$\cap$$ Z = $$\varphi$$

$$\Rightarrow$$ Total ways = 3 × 3 × 3 × 3 × 3 [$$\because$$ no. of elements in set X = 5]

= 3⁵