JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Probability

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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) $\frac{129}{182}$ ✓ Correct
(B) $\frac{103}{182}$
(C) $\frac{17}{26}$
(D) $\frac{19}{26}$

Answer: (A)

Solution

A committee of

12

is chosen from

4

engineers,

2

doctors and

10

professors, and the full selection can be made in

(12 16) = 1820

ways. Requiring at least

3

engineers and at least

1

doctor leaves four admissible compositions:

3 E, 1 D, 8 P

(3 4) (1 2) (8 10) = 360

ways;

3 E, 2 D, 7 P

(3 4) (2 2) (7 10) = 480

ways;

4 E, 1 D, 7 P

(4 4) (1 2) (7 10) = 240

ways; and

4 E, 2 D, 6 P

(4 4) (2 2) (6 10) = 210

ways. These give

1290

favourable committees, so the probability is

\frac{1290}{1820} = \frac{129}{182}

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Single correct option

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