Binomial Theorem
Properties of Binomial Coefficients
Grade 11
Question:
<p>If the rth term in the expansion of \((1 + x)^{20}\) has its coefficient equal to that of the \((r + 4)\)th term, then \(r\) is</p>
<p>(a) 7</p>
<p>(b) 9</p>
<p>(c) 11</p>
<p>(d) 13</p>
Step-by-Step Solution
Key Concept: Use the property that if $\binom{n}{a} = \binom{n}{b}$, then either $a = b$ or $a + b = n$.
<p><strong>Solution:</strong> In the expansion of $(1 + x)^{20}$, the coefficient of the $r$th term is $\binom{20}{r-1}$ and the coefficient of the $(r+4)$th term is $\binom{20}{r+3}$.</p><p>Given: $\binom{20}{r-1} = \binom{20}{r+3}$</p><p>Using the property that $\binom{n}{a} = \binom{n}{b}$ implies $a = b$ or $a + b = n$:</p><p>Case 1: $r - 1 = r + 3$ (impossible)</p><p>Case 2: $(r-1) + (r+3) = 20$</p><p>$2r + 2 = 20$</p><p>$r = 9$</p><p>∴ Answer is (b) 9.</p>
Correct Answer: a