Worksheet generator

1.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.

2.A — 3KA-0076Casework

• BI included 0 and 7 as a pair, even though both numbers have to be bigger than zero.
• CI counted both orders of each pair, even though swapping does not make a new way.
• DI counted every starting number from 1 to 7 without checking which pairs repeat.
• EI counted both orders and also allowed pairs of equal numbers.

3.A — 4KA-0154Inclusion–exclusion with two sets

• BI found how many students play at least one of the two and answered with that instead.
• CI added 18 and 15 to get 33, subtracted the 30 in the class, and used the 3 left over.
• DI removed the 7 from both groups and subtracted 11 and 8 from 30, taking the overlap away twice.
• EI answered with the number who play both, which is what the question gave me rather than what it asked for.

4.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.

5.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.

6.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.

7.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.

8.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.

9.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.

10.A — 54KA-0177Overcount, then correct

• BI counted every line joining two vertices and forgot to remove the 12 sides.
• CI counted each diagonal from both of its endpoints and forgot to halve.
• DI used 12 x 10 without halving, double counting every diagonal.
• EI subtracted 24 rather than 12, removing each side twice.