Permutations & Combinations
Arrangements with conditions
Grade 11
Question:
<p>The number of ways in which we can arrange the 2n students with n boys \(b_1, b_2, \ldots, b_n\) and n girls \(g_1, g_2, \ldots, g_n\) in a line so that all the boys and all the girls stand in increasing order of their age (Assume they all are of different age)</p>
<p>(a) \(({}^nC_0)^2 + ({}^nC_1)^2 + \ldots + ({}^nC_n)^2\)</p>
<p>(b) \({}^{2n}C_n\)</p>
<p>(c) \(2^n\,[1 \cdot 3 \cdot 5 \ldots (2n-1)]/ n!\)</p>
<p>(d) \({}^{2n}C_{n-1}\)</p>
Step-by-Step Solution
Key Concept: When we arrange 2n students such that boys maintain their relative age order AND girls maintain their relative age order independently, we only need to choose which 2n positions the n boys occupy—the remaining n positions automatically go to girls, and their orders are fixed. This reduces to C(2n,n) rather than (2n)!.
<p><strong>Step 1:</strong> Recognize that we have 2n positions in a line. Boys must appear in increasing order of age (b₁ before b₂ before ... before bₙ), and girls must appear in increasing order of age (g₁ before g₂ before ... before gₙ).</p><p><strong>Step 2:</strong> The age orders of boys and girls are FIXED—there's exactly one way to arrange boys in increasing order and exactly one way to arrange girls in increasing order.</p><p><strong>Step 3:</strong> The only choice we have is: which n positions (out of 2n total positions) should the boys occupy? Once we select n positions for boys, the remaining n positions are for girls.</p><p><strong>Step 4:</strong> The number of ways to choose n positions out of 2n positions for boys is C(2n, n) = 2n!/(n!·n!).</p><p><strong>Step 5:</strong> For each such selection, boys fill their positions in increasing age order (1 way) and girls fill their positions in increasing age order (1 way).</p><p><strong>Verification:</strong> Example with n=1: We have b₁, g₁. Arrangements are: (b₁,g₁) and (g₁,b₁). Count = C(2,1) = 2 ✓</p><p>∴ Answer: <strong>C(2n, n) = 2n!/(n!)²</strong> or <strong>Option B</strong></p>
Correct Answer: B