Question

The number of combinations of $2n$ things taken $n$ at a time when $n$ of the $2n$ things are alike and the rest different is

• A. ${}^2{}^n{C}_0+{}^2{}^n{C}_1+{}^2{}^n{C}_2+.....+{}^2{}^n{C}_n$
• B. $\frac{\left(2n\right)!}{n!}$
• C. $\frac{\left(2n\right)!}{(n!{)}^2}$
• D. $2^n$

Answer 1

Answer: (D)

$2^n$

We have $n$ things which are alike and $n$ different things, making a total of $2n$ things.

At a time, we want to take $n$ things, and so the options would be

$0$ from the first set $n$ from the second set,

$1$ from the first set $(n-1)$ from the second set and so on.

So the number of combinations would be ${}^n{C}_0\times 1{+}^n{C}_1\times 1{+}^n{C}_2\times 1+....{+}^n{C}_n\times 1$ since the number of combinations of picking up alike objects is only $1$ in all the cases.

The expression sums up to $2^n$