Problem 10 · Balance a product · 5 points
Made by AI and not reviewed by a real person.
Choose any 20 students from this class of 40, then list every possible three-person team with one bus rider, one walker, and one cyclist. Students may appear on several teams. What is the greatest possible number of teams on the list?
- 343
- 280
- 288
- 294
- 250
Show a hint
- Think about what happens to the list if you swap a chosen student for an unchosen student who travels differently.
- Let b, w, and c be the chosen numbers from each travel group. Count the possible teams, then compare what happens when you move one place from a larger group to a smaller one.
Show the answer and solution
Answer: D — 294
The chart gives 20 bus riders, 10 walkers, and 10 cyclists. If the chosen group has b, w, and c students of each type, then b+w+c=20 and the list has bwc teams. When two of these numbers differ by at least 2, moving one place from the larger to the smaller increases their product, so the greatest product uses 7, 7, and 6. Each of those counts is available in the class, giving 7×7×6=294 teams. The tempting 250 comes from choosing 10 bus riders, 5 walkers, and 5 cyclists to match the class’s proportions, but the chosen group does not have to keep those proportions.