Permutations & Combinations
Combinatorial counting
Grade 11
Question:
<p>In a plane, there are two families of lines \(y = x + r\), \(y = -x + r\), where \(r \in \{0, 1, 2, 3, 4\}\). The number of squares of diagonals of the length \(\sqrt{2}\) formed by the lines is</p>
<p>(a) 9</p>
<p>(b) 16</p>
<p>(c) \(\frac{3}{2} \cdot C_2\)</p>
<p>(d) \(5C_2 + P_2\)</p>
Step-by-Step Solution
Key Concept: Identify pairs of lines from each family that form squares with diagonal length √2, then count the combinations.
<p>There are two sets of five parallel lines at equal distances. Clearly, lines like $l_1, l_3, m_1$ and $m_3$ form a square whose diagonal's length is $\sqrt{2}$.</p><p>∴ The number of required squares $= 3 \times 3 = 9 = \frac{3}{2} \cdot C_2$</p><p>[Q choices are $(l_1, l_2), (l_2, l_4)$ and $(l_3, l_5)$ for one set, etc.]</p>
Correct Answer: a, c