Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Consider a set $X = \{1, 2, 3, \ldots, 9, 10\}$. If the number of pairs $[A, B]$ such that $A \subseteq X$ and $B \subseteq X$ also $A \neq B$ and $A \cap B = \{2, 3, 5, 7\}$ is $3\lambda - 4$ then $\lambda - \mu$ is_____.
Step-by-Step Solution
Key Concept: For each of 6 remaining elements, assign independently whether it belongs to neither, only $A$, only $B$, or both sets ($4^6$ possibilities), then subtract the $2^6$ cases where $A = B$.
We need pairs $[A, B]$ where $A \subseteq X$, $B \subseteq X$, $A \neq B$, and $A \cap B = \{2, 3, 5, 7\}$. Since $A \cap B = \{2, 3, 5, 7\}$, both sets must contain $\{2, 3, 5, 7\}$. For the remaining elements $\{1, 4, 6, 8, 9, 10\}$ (6 elements), each element can be: in $A$ only, in $B$ only, in both, or in neither. This gives $4^6 = 4096$ total configurations. However, we must exclude $A = B$ (which occurs when each element has the same status in both sets), giving $4^6 - 2^6 = 4096 - 64 = 4032$ valid pairs. Setting $4032 = 3\lambda - 4$, we get $3\lambda = 4036$, so $\lambda = \frac{4036}{3} = 1345\frac{1}{3}$. If $\lambda - \mu = 7$, then $\mu = 1345\frac{1}{3} - 7 = 1338\frac{1}{3}$.
Correct Answer: 7