Question

Number of ways of forming a team consisting of atleast $2$ and atmost $(n-2)$ students from $n$ students is

• A. $2^n-2n-2$
• B. $2^n-2n-4$
• C. $2^n-2n$
• D. $2^n-2n-3$

Answer 1

The correct option is A $2^n-2n-2$

As team should consist at least $2$ students and at most $(n-2)$ out of $n$ students

So, required number of ways$=2studentsor3students.........(n-3)studentor(n-2)studenst$

Number of ways${=}^n{C}_2{+}^n{C}_3{+}^n{C}_4.......{.}^n{C}_{n-3}{+}^{n}C(n-2)......\left(i\right)$

${}^n{C}_2{+}^n{C}_3+......{+}^n{C}_{n-2}=2^n{-}^n{C}_0{-}^n{C}_1{-}^n{C}_{n-1}{-}^n{C}_n$

${}^n{C}_2{+}^n{C}_3+......{+}^n{C}_{n-2}=2^n-1-n-n-1$

${}^n{C}_2{+}^n{C}_3+......{+}^n{C}_{n-2}=2^n-2n-\mathrm{2........}\left(ii\right)$

Substitute (ii) in (i) , we get

Number of ways$=2^n-2n-2$