AMC 10 · 2008 · #1

Grade 4 counting

Archetype Branching State Enumeration └ Sub-archetype Numeric Operation Tree

interval-arithmeticsequences-arithmetic systematic-enumeration ↑ Prerequisites: mental-arithmetic

📏 Medium solution 💡 1 insight

Problem

A player scores exactly 5 baskets, and each basket is worth either 2 or 3 points. Counting every way the baskets could split between 2-pointers and 3-pointers, how many different total point values are possible?

Pick an answer.

(A)

$\ 2$

(B)

$\ 3$

(C)

$\ 4$

(D)

$\ 5$

(E)

$\ 6$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

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.

1STEP 1

List the six possible splits

Only the count of 3-pointers can change: 0, 1, 2, 3, 4, or 5, with the rest worth 2 points — six splits in all.

(3-pt, 2-pt) = (0,5),(1,4),(2,3),(3,2),(4,1),(5,0)

2STEP 2

Add up each split's total

Multiply and add each split: 10, 11, 12, 13, 14, 15 — for instance 2 threes and 3 twos gives 6+6=12.

10, 11, 12, 13, 14, 15

3STEP 3

Check the extremes

All 2-pointers gives the smallest total, 5·2=10; all 3-pointers the largest, 5·3=15 — the two ends of the list.

min=5·2=10, max=5·3=15

4STEP 4

Count the distinct totals

Each swap raises the total by exactly 1, so it runs unbroken from 10 to 15: 6 distinct values, choice (E).

10,11,12,13,14,15 → 6 values → (E)

Answer

The totals run from 10 to 15 with no gaps, and the count of whole numbers from 10 to 15 inclusive is 15-10+1=6, which agrees with the six listed totals. The answer 6 is exactly the number of baskets plus one, which makes sense: there are 6 choices for how many baskets are 3-pointers (0 through 5), and each choice gives a different total. Smaller choices like 5 come from forgetting one of the endpoint cases.

💡Key takeaway

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

List the six possible splits

Add up each split's total

Check the extremes

Count the distinct totals

More like this