Topics / Logic, Statistics & Misc

Sets, Estimation & Miscellaneous

Set problems not primarily counting, estimation, pattern recognition, unit conversion

Sets, Estimation & MiscellaneousAMC 10 Step by Step

Name: Date: September 13, 2026

Logic, Statistics & Misc · 12 problems have this as their main idea

primary-topic problems (0.9% of all)

What you need to know

- Two sets: $|A \cup B| = |A| + |B| - |A \cap B|$ . Three sets:
$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|.$
- "Exactly one" and "exactly two" counts come from the Venn diagram regions, not from the union formula.
- In a universe of size $N$ , $|A| + |B| - N \le |A \cap B| \le \min(|A|, |B|)$ .
- Packing problems need a lower bound from items that cannot share a container plus a matching construction.

How AMC 10 tests it

Survey problems (positions 3–12): two or three activities with overlaps and a "none" group; one region is asked for.
"At least how many do both" questions: the tool is the bound $|A| + |B| - N$ .
Packing puzzles (positions 8–15): the trap is dividing total size by capacity, ignoring that items cannot be split.
Estimation problems (positions 1–10): round each factor to one or two significant figures.

Standard approaches

Draw the Venn diagram, label the innermost region $x$ , and fill outward.
Apply inclusion-exclusion with $|A \cup B \cup C| = N - (\text{none})$ .
For extremal overlaps, decide which bound the question wants.
For packing, list which combinations fit in one container, bound, then construct.

Worked example

In a class of $40$ students, $24$ play soccer, $20$ basketball, and $18$ tennis. Ten play soccer and basketball, $8$ soccer and tennis, $7$ basketball and tennis, and $2$ play none. How many play exactly one sport?

(A) $13$ (B) $14$ (C) $15$ (D) $16$ (E) $17$

Solution. Let $x$ play all three. The union has $38$ students, so
$38 = 24 + 20 + 18 - 10 - 8 - 7 + x = 37 + x,$
giving $x = 1$ . The exactly-two regions hold $9, 7, 6$ students, so $23$ play at least two sports and $38 - 23 = 15$ play exactly one. The answer is $\boxed{\textbf{(C)}\ 15}$ .

Pitfalls

Reading " $10$ play soccer and basketball" as excluding those who also play tennis.
Forgetting the "none" group when equating the union to the class size.
Stopping at $\lceil \text{total}/\text{capacity} \rceil$ in packing problems.
Rounding every factor downward in an estimate.

Traps that recur

Forgetting that 1 through 24 are unconstrained and answering 50 (one from each of fifty imagined pairs). (2005 AMC 10B #25)
Assuming the answer is just the 20% net change, or letting more students switch Yes-to-No than the 30% who answer No at the end. (2010 AMC 10B #12)
Dividing total size 16.8 MB by 1.44 to get 12 disks, ignoring that files cannot be split and that 0.8 and 0.7 cannot share a disk. (2002 AMC 10A #11)
Answering 4 from 52/12 = 4.33 rounded down, or 12 from misreading the question as the number of months. (2011 AMC 10B #11)

Problems, easiest first

2023 AMC 10B · #4Sets, Estimation & Miscellaneous

Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?

2019 AMC 10A · #4Sets, Estimation & Miscellaneous

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn $?$

2013 AMC 10B · #2Sets, Estimation & Miscellaneous

Mr. Green measures his rectangular garden by walking two of the sides and finding that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

2011 AMC 10A · #6Sets, Estimation & Miscellaneous

Set $A$ has 20 elements, and set $B$ has 15 elements. What is the smallest possible number of elements in $A \cup B$ , the union of $A$ and $B$ ?

2010 AMC 10B · #3Sets, Estimation & Miscellaneous

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?

2008 AMC 10B · #2Sets, Estimation & Miscellaneous

A $4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? …

2011 AMC 10B · #11Sets, Estimation & Miscellaneous

There are $52$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?

2008 AMC 10B · #12Sets, Estimation & Miscellaneous

Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 …

2007 AMC 10A · #11Sets, Estimation & Miscellaneous

The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

2010 AMC 10B · #12Sets, Estimation & Miscellaneous

At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and …

2002 AMC 10A · #11Sets, Estimation & Miscellaneous

Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?

2005 AMC 10B · #25Sets, Estimation & Miscellaneous

A subset $B$ of the set of integers from $1$ to $100$ , inclusive, has the property that no two elements of $B$ sum to $125$ . What is the maximum possible number of elements in $B$ ?