QA – Placement Quizzes | Permutation and Combination | Question 11

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:

- 4 engineers = 6C4 = 15
- 3 engineers and 1 doctor =
^{6}C_{3}*^{4}C_{1}= 20*4 = 80 - 2 engineers and 2 doctors =
^{6}C_{2}*^{4}C_{2}= 15*6 = 90 - 1 engineer and 3 doctors =
^{6}C_{1}*^{4}C_{3}= 24

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

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