Worksheet generator

1.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.

2.C — 6KA-0067Tree diagrams

• AI counted the puddings only and forgot that the main course is also a choice.
• BI added the number of mains to the number of puddings instead of multiplying them.
• DI used three puddings as well as three mains, without reading how many puddings there are.
• EI doubled the answer, as if the order of the two courses made a different meal.

3.C — 14KA-0009Counting cubes in a stack, including hidden ones

• AI counted only the cubes on the bottom layer and treated the rest as decoration.
• BI counted only the cubes whose faces I could actually see in the drawing.
• DI assumed every layer was a full 3 by 3 square and multiplied 9 by 2 for two layers.
• EI assumed the stack was a solid 3 by 3 by 3 block because it sits in a corner.

4.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.

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.A — 56KA-0163Selections of a few objects

• BI counted ordered selections, so I counted the same three people once for every order they could stand in.
• CI multiplied the 8 members by the 3 places, which counts something quite different from a selection.
• DI divided the 336 ordered selections by 3 instead of by 3 factorial.
• EI chose 2 people rather than 3.

7.A — 6KA-0173Counting up to symmetry

• BI counted every colouring of the four edges and forgot that rotations make some of them identical.
• CI divided 16 by 4, but that only works when no painting is left unchanged by a rotation, and some are.
• DI halved 16, treating only the 180 degree turn as a symmetry.
• EI listed by how many edges are black but forgot that two black edges can be adjacent or opposite.

8.D — 19KA-0018Complementary counting

• AI counted the codes that avoid 3 entirely and gave that as my answer without subtracting.
• BI counted the codes with a 3 in the first position only and forgot the other two positions.
• CI counted the codes with exactly one 3 and forgot the ones with two or three of them.
• EI counted every possible code and forgot to remove the ones with no 3 at all.

9.A — 8KA-0031Counting paths on a grid

• BI assumed closing one junction removes about half the routes and halved the total of 20.
• CI counted the routes that pass through the closed junction and gave that instead of the ones that avoid it.
• DI subtracted only the 2 routes that reach the closed junction, not all the routes that continue through it.
• EI counted every route on the open grid and forgot to remove the ones through the closed junction.

10.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.

11.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.

12.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.

13.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.

14.E — 14KA-0133Counting shapes hidden inside a figure

• AI counted the nine small squares and stopped, missing every larger one.
• BI counted the small squares and the whole board, forgetting the middle size entirely.
• CI found the 2 by 2 squares by splitting the board into blocks, getting two instead of four.
• DI slid the 2 by 2 square across and down but missed one of its four positions.

15.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.