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Product Rule for Counting

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 11 mark

A parking permit code is made of two letters followed by three digits.

Each letter can be any of the 26 letters from A to Z.

Each digit can be any digit from 0 to 9

Letters and digits can be repeated.

How many different codes are possible?

Select the correct answer.

Hint

Count the number of choices for each of the five positions, then multiply (don't add).

Worked solution
  1. There are 26 choices for each letter and 10 choices for each digit.
  2. Repeats are allowed, so the choices don't go down.
  3. Multiply: 26\times26\times10\times10\times10
  4. =676\,000

Question 22 marks

How many odd three-digit integers have three different digits?

Hint

Fill the most restricted places first: the last digit must be odd, and the first digit cannot be 0

Worked solution
  1. Last digit (odd): 1, 3, 5, 7 or 9, so 5 choices.
  2. First digit: not 0 and not the last digit, so 10-2=8 choices.
  3. Middle digit: any digit except the two already used, so 8 choices.
  4. 5\times8\times8=320

Question 3Challenge5 marks

An ice-cream shop sells n different flavours.

A customer orders a cone with three scoops stacked one above another.

The three scoops are all different flavours, and the order of the scoops from bottom to top matters.

There are 2730 different cones possible.

(a)

Work out the value of n.

3 marks

(b)

Two of the flavours are chocolate and vanilla.

How many of the possible cones have chocolate as the bottom scoop and do not contain vanilla?

2 marks

Hint

Write the number of choices for the bottom, middle and top scoop in terms of n, multiply, and set this equal to 2730

Worked solution

Part (a)

  1. Bottom scoop: n choices; middle: n-1; top: n-2
  2. So n(n-1)(n-2)=2730
  3. Three consecutive integers multiply to 2730, so they are close to \sqrt[3]{2730}\approx14.0
  4. Try 13\times14\times15=2730 ✓
  5. So n-2=13 and n=15

Part (b)

  1. Bottom scoop: chocolate only, so 1 choice.
  2. Middle scoop: not chocolate and not vanilla, so 15-2=13 choices.
  3. Top scoop: 12 choices left.
  4. 1\times13\times12=156

More on this topic: Product Rule for Counting worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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