JEE Main 2025 Apr 4 Shift 1, Mathematics Q5: Introduction To Probability

JEE Main2025Apr 4, Shift 1Mathematics

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
$\frac{19}{26}$

Solution

Total ways to pick 12 from $4 + 2 + 10 = 16$ people: $(12 16) = 1820$ .

Enumerate favorable cases by (engineers, doctors, professors):

(E, D, P)	Count
$(3, 1, 8)$	$(3 4) (1 2) (8 10) = 4 \cdot 2 \cdot 45 = 360$
$(3, 2, 7)$	$(3 4) (2 2) (7 10) = 4 \cdot 1 \cdot 120 = 480$
$(4, 1, 7)$	$(4 4) (1 2) (7 10) = 1 \cdot 2 \cdot 120 = 240$
$(4, 2, 6)$	$(4 4) (2 2) (6 10) = 1 \cdot 1 \cdot 210 = 210$
Total favorable	$1290$

Probability $= \frac{1290}{1820} = \frac{129}{182}$ .

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JEE Main 2025 Apr 4 Shift 1, Mathematics Q5: Introduction To Probability

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