Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade 11
Question:
A guardian with $6$ wards wishes everyone of them to study either Law or Medicine or Engineering. Number of ways in which he can make up his mind with regard to the education of his wards if every one of them be fit for any of the branches to study, and atleast one child is to be sent in each discipline is:
Step-by-Step Solution
Key Concept: Use inclusion-exclusion principle to count surjective functions from 6 wards onto 3 disciplines, ensuring each discipline receives at least one ward.
We need to distribute 6 wards among 3 disciplines (Law, Medicine, Engineering) such that each discipline gets at least one ward. Using the inclusion-exclusion principle, the total ways without restriction is $3^6 = 729$. We subtract cases where at least one discipline is empty: $\binom{3}{1} \cdot 2^6 = 3 \times 64 = 192$ (one discipline empty). We add back cases where at least two disciplines are empty: $\binom{3}{2} \cdot 1^6 = 3 \times 1 = 3$ (two disciplines empty). By inclusion-exclusion: $729 - 192 + 3 = 540$.
Correct Answer: 4