AMC 10 · 2008 · #1
Grade 4 countingPick an answer.
The phrase "how many different numbers" is the classic signal for Tool #2 (Make a Systematic List): with only 5 baskets there are just a few splits, so list them all instead of guessing. Tool #14 (Extreme Principle) pins the two ends of the list — all 2-pointers gives the smallest total, all 3-pointers the largest — and Tool #5 (Look for a Pattern) shows the totals climb by exactly 1 each step, so nothing in between is skipped or repeated.
List the six possible splits
With the count fixed, only the split varies, giving six cases.
With the number of baskets fixed, the only choice is how many are 3-pointers, so just list every count from 0 to 5.
4.OA.A.3Make A Systematic ListAdd up each split's total
Each swap raises the total by exactly one.
Each total is just points-from-threes plus points-from-twos.
3.OA.D.8Make A Systematic ListCheck the extremes
The two extremes bracket everything from 10 to 15.
Pushing every basket to its lowest value gives the floor, and to its highest value gives the ceiling.
4.OA.A.3Extreme PrincipleCount the distinct totals
Nothing repeats, so the count is 6, choice (E).
Because each swap adds exactly 1, the totals form an unbroken run, so counting them is just counting from 10 up to 15.
Because each swap adds exactly one point, the totals form an unbroken run with no gaps.
▸ Why?
Trading a two-pointer for a three-pointer changes the total by the same fixed step every time.
▸ Why?
Each count of three-pointers gives its own total and no two counts give the same one, so counting totals is counting splits.
When only one thing can change, list every case in order — here the totals climb from 10 to 15 by ones, so there are 6 of them.
- List the six possible splits
- Add up each split's total
- Check the extremes
- Count the distinct totals