Out of 6 engineers and 4 doctors, how many groups of 4 professionals can be formed such that at least 1 engineer is always there?
(A) 129
(B) 109
(C) 229
(D) 209


Answer: (D)

Explanation: There are four cases:

1. 4 engineers = 6C4 = 15
2. 3 engineers and 1 doctor = ⁶C₃*⁴C₁ = 20*4 = 80
3. 2 engineers and 2 doctors = ⁶C₂*⁴C₂ = 15*6 = 90
4. 1 engineer and 3 doctors = ⁶C₁*⁴C₃ = 24

Therefore, total number of ways = 15 + 80 + 90 + 24 = 209.

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