Question

The probability of forming a 12 persons committee from 4 engineers, 2 doctors, and 10 professors containing at least 3 engineers and at least 1 doctor is:

• A. $\frac{129}{182}$
• B. $\frac{103}{182}$
• C. $\frac{17}{26}$
• D. $\frac{19}{26}$

Hint:

The probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes.

Answer 1

The Correct Option is A

1. Committee formation calculation: - 3 engineers, 1 doctor, 8 professors: \(^4C_3 \cdot ^2C_1 \cdot ^{10}C_8 = 360\) - 3 engineers, 2 doctors, 7 professors: \(^4C_3 \cdot ^2C_2 \cdot ^{10}C_7 = 480\) - 4 engineers, 1 doctor, 7 professors: \(^4C_4 \cdot ^2C_1 \cdot ^{10}C_7 = 240\) - 4 engineers, 2 doctors, 6 professors: \(^4C_4 \cdot ^2C_2 \cdot ^{10}C_6 = 210\) 2. Total favorable outcomes: Total = \(360 + 480 + 240 + 210 = 1290\) 3. Total ways to form a 12-person committee from 16 people: \({}^{16}C_{12} = \frac{16!}{12! \cdot 4!} = 1820\) 4. Probability calculation: Probability = \(\frac{1290}{1820} = \frac{129}{182}\) The final answer is \(\frac{129}{182}\).