Probability
Probability
star_batch_jee_advanced_2025
Grade 12
Question:
The probability so that $r$ $1 \times 1$ squares which are selected from a $m \times n$ chess board such that no two of them share the same row or same column is :
$\frac{^mC_r \cdot ^nC_r}{^{mn}C_r}$
$\frac{^mC_r \cdot ^nC_{r-1}}{^{mn}C_r}$
$\frac{^mC_r \cdot ^nC_r}{^{mn}C_r}$
None of these
Step-by-Step Solution
Key Concept: Select rows and columns independently, then count valid square placements using factorial arrangements.
First select $m$ rows and $r$ columns using $^mC_r$ and $^nC_r$ ways respectively. For any row-column selection, one square occupies position in the first selected row and first selected column, then subsequent rows and columns each contribute one square in $r!$ ways. Total favorable outcomes: $^mC_r \cdot {}^nC_r \cdot r!$. Probability is $\frac{^mC_r \cdot {}^nC_r \cdot r!}{^mC_r}$.
Correct Answer: 4