Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Ramesh has $2n$ number of fruits out of which $n$ of them are identical and remaining $n$ are distinct. If the total number of ways to distribute these fruits to his two children Bhavesh and Sanjesh such that both of them will receive equal number of fruits is 16 then $n$ is equal to ______.
Step-by-Step Solution
Key Concept: Each child receiving $n$ fruits means we distribute $n$ identical fruits (which can be split in $n+1$ ways) and then for each split, choose which distinct fruits each child gets, giving $2^n$ total ways.
We have $2n$ fruits: $n$ identical and $n$ distinct. Each child must receive $n$ fruits. For the $n$ identical fruits, we distribute them such that each child gets some portion - if child 1 gets $k$ identical fruits, child 2 gets $n-k$ identical fruits. For the $n$ distinct fruits, we choose which $m$ distinct fruits go to child 1 (the remaining $n-m$ go to child 2). The total ways is $\sum_{k=0}^{n} \binom{n}{m_k}$ where we pair each distribution of identical fruits with distributions of distinct fruits. Since each child needs exactly $n$ fruits total and gets $k$ identical fruits, they need $n-k$ distinct fruits. The number of ways is $\sum_{k=0}^{n} \binom{n}{n-k} = 2^n$. Setting $2^n = 16 = 2^4$, we get $n = 4$.
Correct Answer: 4