<p>A teacher takes three children from her class to a zoo at a time, but she does not take the same three children to the zoo more than once. She finds that she went to the zoo 84 times more than a particular child has gone to the zoo. The number of children in her class is</p>
Step-by-Step Solution
Key Concept: If a teacher takes 3 children at a time from n children, total trips = C(n,3). A specific child goes in C(n-1,2) trips. Set up the equation C(n,3) = C(n-1,2) + 84 to find n.
<p><strong>Step 1:</strong> Let the total number of children in class = n.</p><p>Total number of trips teacher takes = C(n,3) = n(n-1)(n-2)/6</p><p><strong>Step 2:</strong> A particular child goes to the zoo when they are selected as one of the 3 children. This happens in C(n-1,2) trips (we choose 2 other children from remaining n-1 children).</p><p>C(n-1,2) = (n-1)(n-2)/2</p><p><strong>Step 3:</strong> According to the problem:</p><p>C(n,3) = C(n-1,2) + 84</p><p>n(n-1)(n-2)/6 = (n-1)(n-2)/2 + 84</p><p><strong>Step 4:</strong> Simplify by dividing through by (n-1)(n-2):</p><p>n/6 = 1/2 + 84/[(n-1)(n-2)]</p><p>n/6 - 1/2 = 84/[(n-1)(n-2)]</p><p>(n-3)/6 = 84/[(n-1)(n-2)]</p><p><strong>Step 5:</strong> Cross multiply:</p><p>(n-3)(n-1)(n-2) = 504</p><p><strong>Step 6:</strong> Test values: For n = 9:</p><p>(9-3)(9-1)(9-2) = 6 × 8 × 7 = 336 ✗</p><p>For n = 10:</p><p>(10-3)(10-1)(10-2) = 7 × 9 × 8 = 504 ✓</p><p><strong>Verification:</strong> C(10,3) = 120, C(9,2) = 36, and 120 - 36 = 84 ✓</p><p>∴ Answer: <strong>B (n = 10)</strong></p>
Correct Answer: B