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Math Kangaroo practice

Name: ______________________Date: ____________5 questions · 23 points

1.A large triangle is cut by two lines from its top corner into three small triangles side by side. How many triangles of any size appear in the picture?[4]
Text description of the figure
A large triangle with two straight lines drawn from its top vertex down to the base, dividing it into three small triangles side by side.
- A3
- B4
- C5
- D6
- E9
2.A coin is tossed three times. In how many of the possible outcomes are there exactly two heads?[4]
- A2
- B3
- C4
- D6
- E8
3.Four beads - one red, one green, one blue, one yellow - are threaded on a circular bracelet. Two bracelets are the same if one can be rotated into the other. How many different bracelets are there?[5]
- A3
- B4
- C6
- D12
- E24
4.A code uses the letters A, B and C, each exactly once. How many codes do not start with A?[5]
- A2
- B3
- C4
- D6
- E9
5.An ordinary six-sided die is rolled three times. What is the probability that at least one roll shows a six?[5]
- A91/216
- B1/2
- C125/216
- D3/216
- E1/6

Math Kangaroo Star · https://kangaroo-atlas.vercel.app · seed 9 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.

Answer key — Math Kangaroo practice

Each wrong choice carries the thinking that produces it. When a student picks C, this is what they were doing.

1.D — 6KA-0003Counting shapes hidden inside a figure
- AI counted only the three small triangles I could see as separate cells and stopped there.
- BI counted the three small triangles and added the whole big one, but forgot the two made of two cells.
- CI found the three small ones and one pair, but missed the second pair of neighbouring triangles.
- EI assumed every choice of two of the four lines would make a triangle, without checking each one.
2.B — 3KA-0068Tree diagrams
- AI read the two in exactly two heads as the answer rather than as a condition to count.
- CI counted the outcomes with two or more heads, so I included the three-head one as well.
- DI counted the orderings of three different objects instead of listing the eight outcomes.
- EI gave the total number of possible outcomes rather than the number with exactly two heads.
3.C — 6KA-0025Counting up to symmetry
- AI also treated bracelets that are mirror images as the same, but only rotations were allowed.
- BI divided the number of beads by itself rather than dividing the number of arrangements by the number of rotations.
- DI divided the 24 arrangements by 2 instead of by the 4 rotations of a circle of four beads.
- EI counted every arrangement in a line and forgot that rotating a bracelet gives the same bracelet.
4.C — 4KA-0140The multiplication principle
- AI counted the codes that do start with A rather than the ones that do not.
- BI counted the letters available for the first position instead of counting whole codes.
- DI counted every arrangement of the three letters and forgot the restriction.
- EI allowed letters to repeat, which the words each exactly once rule out.
5.A — 91/216KA-0168Complementary counting
- BI added 1/6 three times, which counts the overlapping cases more than once and would exceed 1 for seven rolls.
- CI worked out the probability of NO six and forgot to subtract it from 1.
- DI found the probability of three sixes instead of at least one.
- EI gave the probability for a single roll and ignored that there are three.

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