Permutations & Combinations
Combinations
Grade 11
Question:
<p>If \(\alpha = {}^mC_2\), then \({}^\alpha C_2\) is equal to</p>
<p>\({}^{m+1}C_4\)</p>
<p>\({}^{m-1}C_2\)</p>
<p>\(3 \cdot {}^{m+2}C_4\)</p>
<p>\(3 \cdot {}^{m+1}C_4\)</p>
Step-by-Step Solution
Key Concept: Recognize that α = mC₂ = m(m-1)/2, then substitute this expression into αC₂ and simplify using the combination formula to get a factorizable form involving m.
<p><strong>Step 1:</strong> Express α in terms of m.</p><p>α = ᵐC₂ = m(m-1)/2</p><p><strong>Step 2:</strong> Find ᵅC₂ using the combination formula.</p><p>ᵅC₂ = α(α-1)/2 = [m(m-1)/2] · [m(m-1)/2 - 1] / 2</p><p><strong>Step 3:</strong> Simplify the numerator.</p><p>α - 1 = m(m-1)/2 - 1 = [m(m-1) - 2]/2 = [m² - m - 2]/2 = (m-2)(m+1)/2</p><p><strong>Step 4:</strong> Substitute back into ᵅC₂.</p><p>ᵅC₂ = [m(m-1)/2] · [(m-2)(m+1)/2] / 2</p><p>ᵅC₂ = m(m-1)(m-2)(m+1) / 8</p><p><strong>Step 5:</strong> Recognize the pattern.</p><p>ᵅC₂ = m(m+1)(m-1)(m-2) / 8 = ᵐC₄ · 4!/2 = <strong>ᵐC₄</strong></p><p>∴ Answer: C</p>
Correct Answer: C