Question

The number of selection of $n$ objects from $2n$ objects of which $n$ are identical and the rest are different is

• A. $2^n$
• B. $2^{n-1}$
• C. $2^n-1$
• D. $2^{n-1}+1$

Answer 1

The correct option is A $2^n$

If all the $n$ objects are chosen from $n$ identical objects, then number of ways is ${}^n{C}_0$
If $n-1$ objects are chosen from $n$ identical objects and $1$ from $n$ diferent objects, then number of ways is ${}^n{C}_1$

Similarly,
If $n-2$ objects are chosen from $n$ identical objects and $1$ from $n$ diferent objects, then number of ways is ${}^n{C}_2$

Therefore, total number of ways is,
${}^n{C}_0+{}^n{C}_1+{}^n{C}_2+...+{}^n{C}_n=2^n$