Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade None
There are $(p + q)$ different books on different topics in Mathematics. $(p \neq q)$
If $L =$ The number of ways in which these books are distributed between two students $X$ and $Y$ such that $X$ get $p$ books and $Y$ gets $q$ books.
$M =$ The number of ways in which these books are distributed between two student $X$ and $Y$ such that one of them gets $p$ books and another gets $q$ books.
$N =$ The number of ways in which these books are divided into groups of $p$ books and $q$ books then:
Step-by-Step Solution
Key Concept: The critical difference is whether the recipients are labeled (students $X$ and $Y$) or unlabeled (just groups), which determines whether to divide by 2 for indistinguishable partitions.
For $L$: Since $X$ must get exactly $p$ books and $Y$ gets $q$ books, we have $L = inom{p+q}{p}$. For $M$: Either $X$ gets $p$ and $Y$ gets $q$, or $X$ gets $q$ and $Y$ gets $p$. Since $p
eq q$, these are different scenarios, so $M = inom{p+q}{p} + inom{p+q}{q} = 2inom{p+q}{p}$. For $N$: When dividing into unlabeled groups, the two groups are indistinguishable, so $N = rac{1}{2}inom{p+q}{p}$ (dividing by 2 to account for overcounting). Therefore: $M = 2L$ and $L = 2N$, giving us $2L = M = 2N$.
Correct Answer: 3