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Math Kangaroo practice

Name: ______________________Date: ____________4 questions · 12 points

1.Eight people meet and everyone shakes hands with everyone else exactly once. How many handshakes take place?[3]
- A16
- B28
- C36
- D56
- E64
2.How many of the numbers from 1 to 20 are not multiples of 3?[3]
- A6
- B13
- C14
- D17
- E20
3.How many different two-digit numbers can be written using only the digits 1, 2 and 3, if a digit may be used twice?[3]
- A3
- B6
- C9
- D12
- E27
4.How many three-digit numbers can be written using only the digits 1 and 2, with repeats allowed?[3]
- A2
- B3
- C4
- D6
- E8

Kangaroo Atlas · https://kangaroo-atlas.vercel.app · seed 1 · 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.B — 28KA-0035Handshakes and pairs
- AI doubled the number of people, as if each person made two handshakes in total.
- CI let each person shake hands with all eight people including themselves before halving.
- DI multiplied 8 by 7 and forgot that each handshake was counted from both sides.
- EI multiplied 8 by 8, counting every ordered pair including a person with themselves.
2.C — 14KA-0051Complementary counting
- AI counted the multiples of 3 and gave that instead of the numbers that are not multiples.
- BI counted 21 as a multiple of 3 inside the range, so I removed one too many.
- DI removed only the multiples of 3 that are also even, so I subtracted three instead of six.
- EI gave the size of the whole range and forgot to remove anything at all.
3.C — 9KA-0072Systematic listing
- AI counted only the numbers with two identical digits, like 11, 22 and 33.
- BI required the two digits to be different, even though repeats are allowed.
- DI allowed a fourth digit that was not in the list.
- EI counted three-digit numbers instead of two-digit ones.
4.E — 8KA-0143Systematic listing
- AI counted the two available digits rather than the numbers they can build.
- BI counted the three positions instead of the numbers.
- CI used only two of the three positions when multiplying.
- DI multiplied 2 by 3, mixing the number of digits with the number of positions.

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