Probability
Probability
Allen Star Batch
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: Selecting r non-attacking squares requires choosing r rows from m rows and r columns from n columns, then multiplying by r! arrangements since each selected row-column pair can be matched in r! ways. The probability is the ratio of favorable outcomes (^mC_r · ^nC_r · r!) to total ways to select r squares from mn squares (^mnC_r).
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