Permutations & Combinations
Grade None
Question:
<p>There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:</p>
<p style="display:inline">2250</p>
<p style="display:inline">2255</p>
<p style="display:inline">1500</p>
<p style="display:inline">3000</p>
Step-by-Step Solution
Key Concept: Identify all valid numerical partitions of the total questions among the sections and sum the selection combinations for each permutation of these partitions.
<p>Since each section has 5 questions.<br />
<span class="math-tex">\(\therefore\)</span> Total number of selection of 5 questions<br />
= 3 <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>1</sub> <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>1</sub> <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>3</sub> + 3 <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>1</sub> <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>2</sub> <span class="math-tex">\(\times\)</span> <sup>5</sup>C<sub>2</sub><br />
= 3 <span class="math-tex">\(\times\)</span> 5 <span class="math-tex">\(\times\)</span> 5 <span class="math-tex">\(\times\)</span> 10 + 3 <span class="math-tex">\(\times\)</span> 5 <span class="math-tex">\(\times\)</span> 10 <span class="math-tex">\(\times\)</span> 10 = 750 + 1500 = 2250</p>
Correct Answer: A