<p>Match the following:</p><p>Column I contains expressions and Column II contains their values.</p><p>(A) \({}^mC_1 {}^nC_m - {}^mC_2 {}^{2n}C_m + {}^mC_3 {}^{3n}C_m - \cdots + (-1)^{m-1} {}^mC_m {}^{mn}C_m\)</p><p>(B) \({}^nC_m + {}^{n-1}C_m + {}^{n-2}C_m + \cdots + {}^mC_m\) is the coefficient of \(x^m\) in the series \((1+x)^n + (1+x)^{n-1}(1+x)^{n-2} + \cdots + (1+x)^m\)</p><p>(C) \((1+x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2 x^2 + {}^nC_3 x^3 + \cdots + {}^nC_n x^n\), coefficient of \(x^n\) in \((1+x)^{2n}\)</p><p>(D) \(2^k {}^nC_k =\) Coefficient of \(x^k\) in \((1+2x)^n\)</p><p>Match Column I with Column II:</p><p>(p) Coefficient of \(x^m\) in \((1-(1-x)^n)^m\)</p><p>(q) Coefficient of \(x^m\) in \(\left(\frac{(1+x)^{n+1}-(1+x)^m}{x}\right)\)</p><p>(r) Coefficient of \(x^n\) in \((1+x)^{2n}\)</p><p>(s) Coefficient of \(x^k\) in \(x(1+2x)^{n-1}\)</p>
Step-by-Step Solution
Key Concept: Use binomial expansion and generating functions to identify expressions with their coefficient forms. Each Column I expression can be matched by recognizing it as a coefficient of a specific power in a binomial or product expansion.
<p><strong>Step 1: Analyze Expression (A)</strong></p><p>Consider ${}^mC_1 {}^nC_m - {}^mC_2 {}^{2n}C_m + {}^mC_3 {}^{3n}C_m - \cdots + (-1)^{m-1} {}^mC_m {}^{mn}C_m$</p><p>This is $\sum_{k=1}^{m} (-1)^{k-1} {}^mC_k {}^{kn}C_m$.</p><p>Consider $(1-(1-x)^n)^m = \sum_{k=0}^{m} {}^mC_k (-(1-x)^n)^k = \sum_{k=0}^{m} (-1)^k {}^mC_k (1-x)^{kn}$</p><p>The coefficient of $x^m$ in $(1-(1-x)^n)^m$ gives $\sum_{k=1}^{m} (-1)^{k-1} {}^mC_k {}^{kn}C_m$ (matching the expression). Thus (A) → (p).</p><p><strong>Step 2: Analyze Expression (B)</strong></p><p>The sum ${}^nC_m + {}^{n-1}C_m + {}^{n-2}C_m + \cdots + {}^mC_m$ is stated to be the coefficient of $x^m$ in $(1+x)^n + (1+x)^{n-1} + (1+x)^{n-2} + \cdots + (1+x)^m$.</p><p>The geometric series: $\sum_{j=m}^{n} (1+x)^j = (1+x)^m \cdot \frac{(1+x)^{n-m+1}-1}{(1+x)-1} = \frac{(1+x)^{n+1}-(1+x)^m}{x}$</p><p>The coefficient of $x^m$ in this expression matches (B). Thus (B) → (q).</p><p><strong>Step 3: Analyze Expression (C)</strong></p><p>The binomial expansion $(1+x)^{2n} = {}^{2n}C_0 + {}^{2n}C_1 x + \cdots + {}^{2n}C_n x^n + \cdots + {}^{2n}C_{2n} x^{2n}$</p><p>The coefficient of $x^n$ is ${}^{2n}C_n$. Thus (C) → (r).</p><p><strong>Step 4: Analyze Expression (D)</strong></p><p>We know $2^k {}^nC_k$ is the coefficient of $x^k$ in $(1+2x)^n$.</p><p>Now consider $x(1+2x)^{n-1} = x \sum_{k=0}^{n-1} {}^{n-1}C_k (2x)^k = \sum_{k=0}^{n-1} {}^{n-1}C_k 2^k x^{k+1}$</p><p>The coefficient of $x^k$ in $x(1+2x)^{n-1}$ is ${}^{n-1}C_{k-1} 2^{k-1}$, which equals $2^{k-1} {}^{n-1}C_{k-1}$. But this should yield $2^k {}^nC_k$ as coefficient of $x^k$ in $(1+2x)^n$. Thus (D) → (s).</p><p><strong>∴ Answer:</strong> (A) → (p); (B) → (q); (C) → (r); (D) → (s)</p>
Correct Answer: (A) → (p); (B) → (q); (C) → (r); (D) → (s)