The number of ways to select n objects from 2 | vedclass

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(A) We need to select $n$ objects from $2n$ objects, where $n$ objects are identical and $n$ objects are distinct.Let $k$ be the number of distinct ob

Vedclass · Accepted Answer

$2^{n}$

Vedclass · Answer

(A) We need to select $n$ objects from $2n$ objects, where $n$ objects are identical and $n$ objects are distinct.Let $k$ be the number of distinct objects selected, where $0 \le k \le n$.Then the remaining $(n-k)$ objects must be selected from the $n$ identical objects.Since the $n$ identical objects are indistinguishable, there is only $1$ way to select any number of them.Thus, for each $k$ from $0$ to $n$, the number of ways to select $k$ distinct objects is given by $\binom{n}{k}$.The total number of ways is the sum of these possibilities:$\sum_{k=0}^{n} \binom{n}{k} = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \dots + \binom{n}{n} = 2^{n}$.

The number of ways to select $n$ objects from $2n$ objects, of which $n$ are identical and the rest are different, is:

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