JEE Main 2024 — 5 April, Shift 1 — Question 21

Let X be the number of defective items in a sample of 5 from 10 items with 3 defectives.

$X \sim$ $(N=10, K=3, n=5)$

$\text{Variance\; of\; } X: \quad \sigma^2 = n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1}$

$\Rightarrow \sigma^2 = 5 \cdot \frac{3}{10} \cdot \frac{7}{10} \cdot \frac{5}{9} = \frac{7}{12}$

$\text{Hence,\; } 96 \sigma^2 = 96 \cdot \frac{7}{12} = 56$

$\boxed{56}$