Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
Suppose a lot contains $2n$ objects of which $n$ are identical. The number of ways to select $n$ objects out of these $2n$ objects must be:
$2^n
$(2^{n+1}C_0 + 2^{n+1}C_1 + \ldots + 2^{n+1}C_n)^{1/2}$
The number of possible subsets of the set $\{a_1, a_2, \ldots, a_n\}$
None of these
Step-by-Step Solution
Key Concept: The number of ways to select $n$ objects when $n$ are identical and $n$ are distinct equals $2^n$, which counts all possible subsets of the distinct objects.
We have $2n$ objects: $n$ identical objects and $n$ distinct objects. When selecting $n$ objects, we choose some number $k$ (where $0 \leq k \leq n$) of identical objects and $(n-k)$ distinct objects from the $n$ distinct ones. This gives $\binom{n}{n-k}$ ways for each $k$. The total is $\sum_{k=0}^{n}\binom{n}{k}=2^n$. This equals the number of subsets of $\{a_1,a_2,\ldots,a_n\}$ (option 3). For option 2: using the identity $\sum_{i=0}^{n}\binom{2n+1}{i}=2^{2n}$, we get $\sqrt{2^{2n}}=2^n$ (option 2).
Correct Answer: 1,2,3