<p>In how many ways can 30 marks be allotted to 8 questions if each question carries at least 2 marks?</p>
Step-by-Step Solution
Key Concept: Convert the constrained problem (minimum 2 marks per question) into an unconstrained one by first allocating 16 marks (2×8), then distributing the remaining 14 marks freely among 8 questions using stars and bars.
<p><strong>Step 1:</strong> Apply the minimum constraint. Since each of 8 questions must carry at least 2 marks, allocate 2×8 = 16 marks first.</p><p><strong>Step 2:</strong> Remaining marks to distribute = 30 - 16 = 14 marks among 8 questions with no restrictions.</p><p><strong>Step 3:</strong> Use the stars and bars formula. The number of ways to distribute n identical items among r recipients is <sup>n+r-1</sup>C<sub>r-1</sub>.</p><p><strong>Step 4:</strong> Apply formula: Distribute 14 marks among 8 questions = <sup>14+8-1</sup>C<sub>8-1</sub> = <sup>21</sup>C<sub>7</sub> = <sup>21</sup>C<sub>14</sub></p><p>∴ Answer: <sup>21</sup>C<sub>14</sub></p>
Correct Answer: \({}^{21}C_{14}\)