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Stage 1 · Combinatorics

10 problems · · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. In the diagram, how many of the shapes are triangles?

   

   1. A$4$
   2. B$5$
   3. C$6$
   4. D$7$
   5. E$8$

   CEMC Gauss (Grade 7) 2026

   Combinatorics Solution and answer checking →

2. A mother bear collects $14$
   fish. She gives $4$ fish to each of
   her $3$ bear cubs. How many fish does
   the mother bear have left over?

   1. A$0$
   2. B$2$
   3. C$3$
   4. D$4$
   5. E$5$

   CEMC Cayley 2025

   Combinatorics Solution and answer checking →

3. A six-sided die has the numbers one to six on its sides. What is the probability of rolling a five?

   1. A$\frac{2}{6}$
   2. B$\frac{1}{6}$
   3. C$\frac{5}{6}$
   4. D$\frac{3}{6}$
   5. E$\frac{4}{6}$

   CEMC Gauss (Grade 8) 2012

   Combinatorics Solution and answer checking →

4. What is the largest number of squares with side length 2 that can
   be arranged, without overlapping, inside a square with side length
   8?

   1. A$8$
   2. B$32$
   3. C$16$
   4. D$64$
   5. E$4$

   CEMC Fermat 2022

   Combinatorics Solution and answer checking →

5. Students at Gauss Middle School were asked to choose their
   favourite school day. The results are shown in the circle graph.

   

   Which day was chosen by exactly one-quarter of the students?

   1. AMonday
   2. BTuesday
   3. CWednesday
   4. DThursday
   5. EFriday

   CEMC Gauss (Grade 7) 2024

   Combinatorics Solution and answer checking →

6. The pattern of shapes $\triangle, \square, \heartsuit, \diamondsuit, \bigcirc$ is repeated to form
   the sequence $$\triangle, \square, \heartsuit, \diamondsuit, \bigcirc, \triangle, \square, \heartsuit, \diamondsuit, \bigcirc, \ldots$$ The 22nd shape in the sequence
   is

   1. A$\square$
   2. B$\bigcirc$
   3. C$\diamondsuit$
   4. D$\triangle$
   5. E$\heartsuit$

   CEMC Cayley 2022

   Combinatorics Solution and answer checking →

7. The graph shows the total distance that each of five runners ran during a one-hour training session.

   

   Which runner ran the median distance?

   1. A$\textrm{Phil}$
   2. B$\textrm{Tom}$
   3. C$\textrm{Pete}$
   4. D$\textrm{Amal}$
   5. E$\textrm{Sanjay}$

   CEMC Gauss (Grade 8) 2015

   Combinatorics Solution and answer checking →

8. The number 2023 is written on one side of a classroom
   window using the style of numbers shown.

   

   If you were to stand on the other side of the window, how would the
   number appear?

   
   
   
   
   

   CEMC Cayley 2023

   Combinatorics Solution and answer checking →

9. Evgeny has 3 roses, 2 tulips, 4 daisies, and 6 lilies. If he randomly chooses one of these flowers, what is the probability that it is a tulip?

   1. A$\frac{3}{15}$
   2. B$\frac{12}{15}$
   3. C$\frac{6}{15}$
   4. D$\frac{4}{15}$
   5. E$\frac{2}{15}$

   CEMC Gauss (Grade 8) 2018

   Combinatorics Solution and answer checking →

10. Carley went for a run on May 18th. She did not go for a run on
    May 19th. She continued to alternate going for a run one day with not
    going for a run the following day. Between May 18th and May 25th
    inclusive, how many days did she go for a run?

    1. A$5$
    2. B$6$
    3. C$3$
    4. D$7$
    5. E$4$

    CEMC Cayley 2026

    Combinatorics Solution and answer checking →

Answer key — Stage 1 · Combinatorics

Worked solutions for every problem are on the site, one page per problem.

1. $C$ open
2. $B$ open
3. $D$ open
4. $C$ open
5. $C$ open
6. $A$ open
7. $B$ open
8. $C$ open
9. $C$ open
10. $E$ open

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