A class has $n$ students, we have to form a team of the students including at least two students and also excluding at least two students. The number of ways of forming the team is

2^{n} - 2 n

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2^{n} - 2 n - 2

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2^{n} - 2 n - 4

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2^{n} - 2 n - 6

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Solution

The correct option is B $2^{n} - 2 n - 2$
As team should consist of at least $2$ & at most $(n - 2)$ students out of n students. So required number of ways $= 2$ students or $3$ students....
or $(n - 3)$ students or $(n - 2)$ students
$\Rightarrow$ number of ways $=^{n} C_{2} +^{n} C_{3} +^{n} C_{4} + . . . . +^{n} C_{n - 3} +^{n} C_{n - 2}$ …….. $(1)$
$^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + . . . . . +^{n} C_{n - 2} +^{n} C_{n - 1} +^{n} C_{n} = 2^{n}$
$\Rightarrow^{n} C_{2} +^{n} C_{3} + . . . . +^{n} C_{n - 2} = 2^{n} -^{n} C_{0} -^{n} C_{1} -^{n} C_{n - 1} -^{n} C_{n}$
$^{n} C_{2} +^{n} C_{3} + . . . . . +^{n} C_{n - 2} = 2^{n} - 1 - n - n - 1$
$^{n} C_{2} +^{n} C_{3} + . . . . . +^{n} C_{n - 2} = 2^{n} - 2 n - 2$ ……… $(2)$
Substituting $(2)$ in $(1)$
Number of ways $= 2^{n} - 2 n - 2$
Hence, the number of ways of forming a team,
$2^{n} - 2 n - 2$ .

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