<p>Find the value of \(({}^{10}C_0) + ({}^{10}C_0 + {}^{10}C_1) + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2) + \ldots + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2 + \ldots + {}^{10}C_9)\).</p>
Step-by-Step Solution
Key Concept: Recognize this as a sum of partial sums of binomial coefficients. Use the identity that the sum of partial sums can be rewritten as ∑(k+1)·ⁿCₖ, which equals (n+2)·2^(n-1) when properly indexed.
<p><strong>Step 1:</strong> Rewrite the sum by counting how many times each term appears:</p><p>⁰C₀ appears 10 times, ¹⁰C₀ appears 9 times, ¹⁰C₁ appears 8 times, ..., ¹⁰C₉ appears 1 time</p><p>Sum = 10·⁰C₀ + 9·¹⁰C₁ + 8·¹⁰C₂ + ... + 1·¹⁰C₉</p><p><strong>Step 2:</strong> Reindex: This equals ∑(k=0 to 9) (10-k)·¹⁰Cₖ = ∑(k=0 to 9) 10·¹⁰Cₖ - ∑(k=0 to 9) k·¹⁰Cₖ</p><p><strong>Step 3:</strong> Apply known identities:</p><p>• ∑(k=0 to 9) ¹⁰Cₖ = 2¹⁰ - ¹⁰C₁₀ = 1024 - 1 = 1023</p><p>• ∑(k=0 to 9) k·¹⁰Cₖ = 10·2⁹ - 10·¹⁰C₁₀ = 10·512 - 0 = 5120</p><p><strong>Step 4:</strong> Combine: Sum = 10(1023) - 5120 = 10230 - 5120 = 5110</p><p>∴ Answer: <strong>5110</strong></p>
Correct Answer: 5110