Permutations & Combinations
Combinations
Grade None
Question:
<p>If \(T_{n+1} - T_n = 10\), then the value of \(n\) is obtained as follows: \({}^{n+1}C_3 - {}^nC_3 = 10 \Rightarrow \dfrac{(n+1)n(n-1)}{6} - \dfrac{n(n-1)(n-2)}{6} = 10 \Rightarrow n(n-1)(n+1-n+2) = 60 \Rightarrow n(n-1) = 20 \Rightarrow n(n-1) = 5 \times 4\). Therefore, \(n =\):</p>
<p>3</p>
<p>4</p>
<p>5</p>
<p>6</p>
Step-by-Step Solution
Key Concept: When finding differences of consecutive binomial coefficients, factor out the common terms n(n-1) and simplify the remaining expression. The key is recognizing that (n+1) - (n-2) = 3 in the numerator, giving you a manageable quadratic.
<p><strong>Step 1:</strong> Write the binomial coefficients using the formula: <sup>n+1</sup>C<sub>3</sub> = (n+1)n(n-1)/6 and <sup>n</sup>C<sub>3</sub> = n(n-1)(n-2)/6</p><p><strong>Step 2:</strong> Find their difference and multiply both sides by 6:</p><p>n(n-1)[(n+1) - (n-2)] = 60</p><p><strong>Step 3:</strong> Simplify the bracket: (n+1) - (n-2) = 3</p><p>n(n-1)(3) = 60</p><p><strong>Step 4:</strong> Divide by 3:</p><p>n(n-1) = 20</p><p><strong>Step 5:</strong> Recognize that 20 = 5 × 4, so n(n-1) = 5 × 4 gives n = 5</p><p><strong>Verification:</strong> 5(4) = 20 ✓</p><p>∴ Answer: n = <strong>5</strong> (Option C)</p>
Correct Answer: C