1. number of ways of forming a team consistiing of atleast 2 and atmost (n-2) students from n sudents is

i) 2n-2n-2

ii) 2n-2n-4

iii) 2n-2n

iv) 2n-2n-3

2. How many non prime factors are in the number N=25*37*92*114*133

Dear Student!

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(1) As team should consist of at least 2 and 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.

⇒ number or ways = ⁿC₂ + ⁿC₃ + ⁿC₄ + ...... ⁿC_{n – 3} + ⁿC_{n – 2}  ...... (I)

ⁿC₀ + ⁿC₁ + ⁿC₂ + ⁿC₃ + ........ + ⁿC_{n – 2} + ⁿC_{n – 1} + ⁿC_n  = 2ⁿ

⇒ ⁿC₂ + ⁿC₃ + ......... + ⁿC_{n – 2}  = 2ⁿ – ⁿC₀ – ⁿC₁ – ⁿC_n–1 –  ⁿC_n 

⇒ ⁿC₂ + ⁿC₃ + ......... + ⁿC_{n – 2}  = 2ⁿ – 1 – n – n – 1

⇒ ⁿC₂ + ⁿC₃ + ......... + ⁿC_{n – 2}  = 2ⁿ – 2n – 2  ......... (II)

Substituting (II) in (I),

Number of ways = 2ⁿ – 2n – 2 

Hence, number of ways of forming a team: (1) 2ⁿ – 2n – 2 

Hope it helps!