1.7 Systematic listing and the product rule for counting
9 exam questions worth 21 marks. Write your answer, open the mark scheme, then give yourself the marks you earned. Your answers are saved on this device.
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Digits 1, 2 and 3 can be repeated. How many two-digit numbers can be made?
Choose one answer.
Correct Not quite: the answer is 9
Mark scheme
C: 9
Model answer
9 (C). 3 choices for the tens digit and 3 for the units digit gives 9.
A coin is flipped twice. List all the possible outcomes.
Mark scheme
M1: attempt to list; A1: HH, HT, TH, TT
Model answer
HH, HT, TH, TT. (H = head, T = tail.)
A café sells two wraps (chicken, falafel) and three drinks (tea, juice, water). List all the possible combinations of one wrap and one drink.
Mark scheme
M1: systematic attempt; A1: at least 5 correct without repeats; A1: all 6 correct: chicken + tea, chicken + juice, chicken + water, falafel + tea, falafel + juice, falafel + water
Model answer
Fix the wrap: chicken with tea, juice or water; then falafel with tea, juice or water. There are 6 combinations.
Four digits 4, 5, 6 and 7 are used to make three-digit numbers. No digit can be used more than once. How many different three-digit numbers can be made?
Mark scheme
M1: 4 choices, then 3, then 2; M1: 4 × 3 × 2; A1: 24
Model answer
There are 4 choices for the first digit, 3 for the second and 2 for the third: 4 × 3 × 2 = 24.
A PIN has 4 digits. Each digit is chosen from 0 to 9 and digits can repeat. How many different PINs are there?
Choose one answer.
Correct Not quite: the answer is 10 000
Mark scheme
C: 10 000
Model answer
10 000 (C). 10 × 10 × 10 × 10 = 10 000.
A meal deal has 3 sandwiches, 4 drinks and 2 snacks. How many different meal deals are possible?
Mark scheme
M1: 3 × 4 × 2; A1: 24
Model answer
3 × 4 × 2 = 24.
A code is made from 2 letters (chosen from A to Z, repeats allowed) followed by 3 digits (0 to 9, repeats allowed). How many different codes are possible?
Mark scheme
M1: 26 × 26 for the letters; M1: × 10 × 10 × 10; A1: 676 000
Model answer
26 × 26 × 10 × 10 × 10 = 676 × 1000 = 676 000.
Spinner A is numbered 1, 2, 3 and spinner B is numbered 1, 2, 3, 4. Both are spun and the scores are added. (a) How many different outcomes are there? (b) How many of these give a total of 5?
Mark scheme
B1: 12; M1: lists pairs adding to 5; A1: 3 ((1,4), (2,3), (3,2))
Model answer
(a) 3 × 4 = 12 outcomes. (b) The pairs (1, 4), (2, 3) and (3, 2) each total 5, so there are 3.
Five friends take part in a race. How many different ways can the first three places be awarded? Explain your method.
Mark scheme
B1: 5 choices for 1st, 4 for 2nd and 3 for 3rd; M1: 5 × 4 × 3; A1: 60
Model answer
There are 5 possible winners. Once someone has won, 4 friends are left for 2nd place, and then 3 are left for 3rd place. 5 × 4 × 3 = 60 ways.
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