GCSE Systematic listing Practice Questions

Free GCSE Systematic listing practice questions with full step-by-step worked solutions. Covers listing outcomes, systematic listing, arrangements, listing combinations. Practise exam-style problems and check your method.

listing outcomessystematic listingarrangementslisting combinationscombinationssample space
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
Two coins are tossed. List all the possible outcomes.
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Worked solution

  1. Fix the first coin as Heads

    HH, HTHH,\ HT

    Keep the first coin on Heads and list the second coin's options: Heads then Tails.

  2. Fix the first coin as Tails

    TH, TTTH,\ TT

    Now the first coin is Tails, and again list the second coin's options.

  3. List all outcomes

    HH, HT, TH, TTHH,\ HT,\ TH,\ TT

    Working systematically gives all four outcomes with none missed or repeated.

Answer
HH,HT,TH,TTHH, HT, TH, TT
Question 2
1 markeasy
An ice cream has one of 33 flavours, served in a cone or a tub. List all the possible ice creams.
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Worked solution

  1. Each flavour comes in 22 containers

    3 flavours×23\ \text{flavours}\times2

    For each of the 33 flavours there are 22 container choices (cone or tub).

  2. Multiply

    3×23\times2

    Use the product rule.

  3. State the total

    66

    So there are 66 possible ice creams.

Answer
66
Question 3
2 marksintermediate
A quiz has 33 true-or-false questions. How many different patterns of answers are possible?
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Worked solution

  1. Each question has 22 answers

    T or FT\ \text{or}\ F

    Every question can be answered true or false.

  2. There are 33 questions

    3 questions3\ \text{questions}

    Each with 22 options.

  3. Product rule

    2×2×22\times2\times2

    Multiply 22 for each question.

  4. Write as a power

    232^3

    Three 22s multiplied is 232^3.

  5. Work it out

    23=82^3=8

    So there are 88 patterns.

  6. State the answer

    88

    There are 88 different answer patterns.

Answer
88
Question 4
3 markshard
How many three-digit numbers, made from the digits 11-99 with digits allowed to repeat, are odd?
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Worked solution

  1. Digits are 1−91-9 with repeats allowed

    1-91\text{-}9

    Each position can be any digit from 11 to 99.

  2. Choices for the hundreds digit

    99

    Any of 1−91-9, so 99 choices.

  3. Choices for the tens digit

    99

    Repeats are allowed, so still 99 choices.

  4. The units digit must be odd

    odd\text{odd}

    For an odd number, the last digit is odd.

  5. Odd digits from 1−91-9

    1,3,5,7,91,3,5,7,9

    There are 55 odd digits.

  6. Choices for the units digit

    55

    So 55 choices for the units.

  7. Product rule

    9×9×59\times9\times5

    Multiply the choices at each position.

  8. First multiplication

    9×9=819\times9=81

    8181 so far.

  9. Multiply by 55

    81×5=40581\times5=405

    So the total is 405405.

  10. State the answer

    405405

    So there are 405405 three-digit odd numbers.

Answer
405405
Question 5
5 markschallenging
A coin is tossed and a dice is rolled. Show that listing the outcomes and using the product rule both give the same number of outcomes.
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Worked solution

  1. A coin has 22 outcomes

    H, TH,\ T

    Heads or Tails.

  2. A dice has 66 outcomes

    1,2,3,4,5,61,2,3,4,5,6

    The numbers 11 to 66.

  3. List method: pair each coin with each dice

    systematic listing\text{systematic listing}

    Keep the coin fixed and run through the dice.

  4. Coin Heads

    H1,H2,H3,H4,H5,H6H1,H2,H3,H4,H5,H6

    Six outcomes with Heads.

  5. Count these

    66

    That is 66 outcomes.

  6. Coin Tails

    T1,T2,T3,T4,T5,T6T1,T2,T3,T4,T5,T6

    Six outcomes with Tails.

  7. Count these

    66

    Another 66 outcomes.

  8. Total from listing

    6+6=126+6=12

    So the list has 1212 outcomes.

  9. Now the product rule

    2×62\times6

    Multiply the coin outcomes by the dice outcomes.

  10. Work it out

    2×6=122\times6=12

    The product rule gives 1212.

  11. Compare the two

    12=1212=12

    Both methods give 1212.

  12. Why they agree

    2 rows of 62\ \text{rows of }6

    Each of the 22 coin results has 66 dice partners.

  13. That is exactly multiplication

    2×62\times6

    Two rows of six is 2×62 \times 6.

  14. So the product rule is a shortcut for listing

    same total\text{same total}

    The product rule counts the same outcomes without writing them all out.

  15. State the conclusion

    1212

    So listing and the product rule both give 1212 outcomes.

Answer
Listing gives 1212 outcomes (H1..H6,T1..T6H1..H6, T1..T6); the product rule gives 2×6=122 \times 6 = 12. Both give 1212.

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