Worksheet generator

1.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.

2.D — 75KA-0023The multiplication principle

• AI added the number of options at each step instead of multiplying them.
• BI treated the two digits as one choice of ten rather than two independent choices of five.
• CI assumed the two digits had to be different, so I used 5 times 4 for them.
• EI used 5 choices for the letter position as well, forgetting there are only three letters.

3.B — 4KA-0069Inclusion–exclusion with two sets

• AI added 18 and 14 without removing the overlap, then subtracted from 30.
• CI subtracted the 6 who play both twice instead of once.
• DI subtracted only the footballers from the class and forgot the chess players entirely.
• EI found how many play at least one sport and gave that instead of how many play neither.

4.D — 24KA-0073Arrangements of a few objects

• AI counted the letters rather than their arrangements.
• BI multiplied 4 by 3 and stopped, forgetting the last two positions.
• CI used 4 choices for every position, as if letters could repeat.
• EI used 4 choices in each of four positions, which allows every letter to repeat.

5.B — 15KA-0074Selections of a few objects

• AI doubled the group size instead of counting pairs.
• CI allowed a student to be paired with themselves, adding six impossible pairs.
• DI counted each pair twice, once in each order, and forgot to halve.
• EI multiplied 6 by 6, counting every ordered pair including a student with themselves.

6.C — 48KA-0070Counting with restrictions

• AI treated the twins as one child and forgot they can also swap places with each other.
• BI glued the twins together into a single block but then forgot that the block can be arranged with the others in more ways than I counted.
• DI halved the 120 total, assuming the twins are together in exactly half the arrangements.
• EI counted every arrangement of five children and forgot the twins' condition entirely.

7.A — 12KA-0071Overcount, then correct

• BI divided the 120 arrangements by 6 rather than by the 10 movements that leave a bracelet looking the same.
• CI allowed for turning the bracelet but forgot it can also be flipped over.
• DI halved the 120 for flipping but forgot that turning also gives the same bracelet.
• EI counted every arrangement in a line, as if the bracelet had a fixed first bead.

8.B — 252KA-0075Complementary counting

• AI counted the numbers made entirely of digits other than 7 in every position, including a leading zero.
• CI counted the numbers with a 7 in each position separately and forgot that some were counted twice.
• DI added three lots of 90 and one extra hundred, double counting the 700s.
• EI counted the numbers with no 7 at all and gave that instead of subtracting it.

9.C — 7KA-0077The pigeonhole principle

• AI answered with the luckiest case, where the first three balls happen to match.
• BI found the worst case of two of each colour and forgot to take one more.
• DI multiplied the three colours by the three balls I need, rather than thinking about the worst case.
• EI worked from the number of balls of each colour rather than from the number of colours.

10.C — 55KA-0078Extremal / worst-case counting

• AI found the smallest possible total of the other nine and gave that instead of what is left.
• BI assumed the biggest number could be at most half the total.
• DI made the other nine numbers all equal to 1, forgetting that they have to be different from each other.
• EI ignored the other nine numbers entirely, as if the biggest could take the whole total.