Worksheet generator

1.C — 8KA-0012Casework

• AI counted each pair of digits once instead of counting both orders, such as 17 and 71.
• BI listed the pairs starting from 1 and 7 and forgot the number 80, where the second digit is zero.
• DI included 08 as a two-digit number, but a two-digit number cannot start with zero.
• EI counted both orders of every pair and then counted the pairs that reverse to themselves twice as well.

2.C — 4KA-0040The pigeonhole principle

• AI answered with the lucky case, where the first two socks happen to match.
• BI used the number of colours as my answer without adding one for the sock that must repeat.
• DI worked from the number of socks of each colour instead of the number of colours.
• EI took the whole drawer, guaranteeing a pair but far more socks than are needed.

3.D — 30KA-0047Counting by position (sliding window)

• AI counted only the sixteen smallest squares and stopped there.
• BI counted the small squares and the single big one, and forgot every size in between.
• CI counted the 2 by 2 squares as four rather than nine, splitting the board into blocks.
• EI used 4 by 4 positions for every size instead of shrinking the range as the square grows.

4.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.

5.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.

6.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.

7.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.

8.B — 7KA-0157The pigeonhole principle

• AI thought that one more than the number of colours must give three of a kind, which only guarantees a PAIR.
• CI got to two of each colour, then added one more for each colour instead of one more in total.
• DI assumed the worst case meant emptying the largest pile of 12 reds first.
• EI answered with the number of socks I want rather than the number I must take to be sure of them.

9.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.

10.A — 8KA-0175The pigeonhole principle

• BI used the number of possible remainders without adding one for the pair that must collide.
• CI doubled 7, thinking I needed two full sets of remainders.
• DI doubled 7 and added one, applying the pigeonhole idea to the wrong number of boxes.
• EI guessed from small cases without identifying what the boxes actually are.