A youth organization created several teams, each consisting of three members.
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A youth organization created several teams, each consisting of three members. Each team is required to sell cookie boxes. Which of the statements is sufficient to determine if at least one member of a certain team sold at least 3 cookie boxes yesterday?
- Statement 1: The three members sold a total of 6 cookie boxes yesterday.
- Statement 2: No two members sold the same number of cookie boxes yesterday.
(Note: Each member can only sell a non-negative whole number of boxes.)
Show answer & explanation
Statement 1 allows for distributions where no one sells 3 boxes (e.g., 2, 2, 2). Statement 2 allows for distributions where no one sells 3 boxes (e.g., 0, 1, 2). However, combining both forces a distribution where at least one member must sell 3 or more boxes to reach the total of 6 while maintaining distinct values.
Step-by-step Derivation:
Step 1: Analyze Statement 1. The total is 6 boxes for 3 members. Possible distributions include (2, 2, 2), where no member sold at least 3 boxes. Thus, Statement 1 is NOT sufficient.
Step 2: Analyze Statement 2. No two members sold the same number. Possible distributions include (0, 1, 2), where no member sold at least 3 boxes. Thus, Statement 2 is NOT sufficient.
Step 3: Combine Statement 1 and Statement 2. We need three distinct non-negative integers (a, b, c) such that a + b + c = 6. Let's list the possible sets of distinct non-negative integers that sum to 6:
- {0, 1, 5} -> Max is 5 (>= 3)
- {0, 2, 4} -> Max is 4 (>= 3)
- {1, 2, 3} -> Max is 3 (>= 3)
Step 4: In every possible scenario where the sum is 6 and all values are distinct, the maximum value is always at least 3. Therefore, both statements together are sufficient.