Permutations & Combinations
Counting with conditions
Grade 11
Question:
<p>A class contains three girls and four boys. Every Saturday, five go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is</p>
<p>21</p>
<p>45</p>
<p>27</p>
<p>24</p>
Step-by-Step Solution
Key Concept: Each girl receives one doll per picnic group she attends. Count how many groups each girl participates in across all possible groups of 5 from 7 students, then sum the dolls.
<p><strong>Step 1:</strong> Find total number of possible groups of 5 from 7 students: C(7,5) = 21 groups</p><p><strong>Step 2:</strong> Use symmetry principle. For any specific girl, she is included in groups where 4 more students are chosen from the remaining 6 students: C(6,4) = 15 groups</p><p><strong>Step 3:</strong> Each girl appears in exactly 15 different picnic groups, so each girl receives 15 dolls.</p><p><strong>Step 4:</strong> There are 3 girls total, so total dolls = 3 × 15 = 45 dolls</p><p><strong>Verification:</strong> Alternatively, in each group of 5, there are on average 3 × (5/7) = 15/7 girls. Total dolls = 21 × 15/7 = 45 ✓</p><p>∴ <strong>Answer: 45</strong></p>
Correct Answer: B