AMC 10 · 2013 · #12

Grade 7 probability

Archetype Casework by Position / Vertex Type └ Sub-archetype Count of Type

probability-basiccombinations-basicregular-pentagon systematic-enumerationcomplementary-counting ↑ Prerequisites: probability-basiccombinations-basic

📏 Medium solution 💡 2 insights

Problem

A regular pentagon has 5 sides and 5 diagonals, giving a set S of 10 segments in all. Two of these segments are drawn at random, without replacement. Find the probability that the two drawn segments are the same length.

Pick an answer.

(A)

$\frac{2}5$

(B)

$\frac{4}9$

(C)

$\frac{1}2$

(D)

$\frac{5}9$

(E)

$\frac{4}5$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

Every pair of the 10 segments is equally likely, so the probability is just (favorable pairs)/(total pairs). Counting pairs is a 'how many ways' job, so I count with combinations. Sketching the pentagon first shows why the lengths split into exactly two equal groups of 5. As a check I also change focus: fix the first pick and ask how many of the remaining 9 match it.

1STEP 1

Two equal groups of five

Draw the pentagon: its 5 sides are equal and its 5 diagonals are equal, so S splits into exactly two length groups of 5.

|S| = 5 sides + 5 diagonals = 10

2STEP 2

Count all possible pairs

Order does not matter, so picking 2 of the 10 segments is 10 choose 2, giving 45 possible pairs.

C(10, 2) = (10 × 9)/2 = 45

3STEP 3

Count the matching pairs

A match needs both from one group, so add 5 choose 2 sides and 5 choose 2 diagonals: 20 pairs.

C(5, 2) + C(5, 2) = 10 + 10 = 20

4STEP 4

Divide to get the probability

Favorable over total gives $\frac{20}{45}$ , which reduces to $\frac{4}{9}$ — choice (B).

P = 20/45 = 4/9

Answer

4/9

4/9 is just under 1/2, which fits: after your first segment is fixed, only 4 of the other 9 segments share its length, and 4 out of 9 is a bit below half. The tempting wrong answer (C) 1/2 comes from mistakenly assuming exactly half the leftovers match. Since 4/9 ＜ 1/2, choice (C) is the trap and (B) is right.

💡Key takeaway

The 10 segments split into two equal groups of 5, so a match means both come from the same group — count those pairs over all pairs and you get 4/9.

Two equal groups of five
Count all possible pairs
Count the matching pairs
Divide to get the probability

Two equal groups of five

Count all possible pairs

Count the matching pairs

Divide to get the probability

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