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Counting and enumerationGrades 1–6

Enumeration

How many numbers can be written with these digits? How many handshakes, outfits or ways to stand in a line? In these tasks every case is listed in an order that makes sure none is lost and none is counted twice. A list kept by a rule is the first tool a child has for questions too big to guess.

The main move
List the cases by a rule
What it teaches
To work in order, and to see when cases can be counted without listing them
      1         2         3
     / \       / \       / \
    2   3     1   3     1   2
   12  13    21  23    31  32

The rules of a list

Does the order matter?

The same three children, A, B and C, give different counts for different questions.

The questionDoes the order matter?The cases
In how many ways can they stand in a line?Yes: AB and BA are two linesABC, ACB, BAC, BCA, CAB, CBA — 6
How many pairs can be sent on an errand?No: AB and BA are one pairAB, AC, BC — 3
In how many ways can a leader and a helper be chosen?Yes: the leader comes firstAB, AC, BA, BC, CA, CB — 6

How to solve: a list by a rule

Make a rule first, what comes first in the list and what next, then write the cases out by it, or count them once the rule shows a pattern. The examples below go from the simplest to an olympiad's.

  1. Decide what makes two cases different
  2. List the cases by a rule
  3. Count them, or multiply when choices are made one after another

Example 1 · Grades 1–2

Flags of two stripes

A flag has two stripes, one above the other. Each stripe is red, yellow or green, and the two stripes are different colours. How many different flags are there?

  1. Put the flags in order by the top stripe. With red on top, the bottom one is yellow or green: two flags.
  2. With yellow on top, red or green below: two more. With green on top, red or yellow: two more.
  3. 3 colours for the top, and each time 2 left for the bottom: 3 × 2 = 6 flags.
   R  R  Y  Y  G  G
   Y  G  R  G  R  Y

Answer 6 flags.

Example 2 · Grades 3–4

Digits three apart

How many two-digit numbers have digits that differ by 3, like 14 or 30?

  1. Go through the tens digit from 1 to 9, and write next to it every units digit 3 away from it: 1 → 4; 2 → 5; 3 → 0 and 6; 4 → 1 and 7; and so on.
  2. With 3, 4, 5 or 6 in the tens, there are two numbers each, 3 below and 3 above. With 1, 2, 7, 8 or 9, just one: the other would not be a digit.
  3. 4 × 2 + 5 × 1 = 13 numbers: 14, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 85, 96.

Answer 13 numbers.

Example 3 · Grades 5–6

No two 2s side by side

How many four-digit numbers can be written with the digits 1 and 2 alone, so that no two 2s stand next to each other?

  1. There are too many to guess, so build them up digit by digit. One digit: 1 and 2, two numbers. Two digits: 11, 12 and 21, three, since 22 is not allowed.
  2. An allowed number ends in 1 or in 2. Ending in 1, it is any allowed number one digit shorter with a 1 added. Ending in 2, the digit before must be 1: it is any allowed number two digits shorter with 12 added. So each count is the sum of the two before it: three digits make 3 + 2 = 5, and four digits 5 + 3 = 8.
  3. Check by listing them: 1111, 1112, 1121, 1211, 2111, 1212, 2112, 2121, eight.

Answer 8 numbers.

Where children go wrong

MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.

How to help at home

  1. Dress up a toy

    Give the child two hats and three scarves for a toy, and ask how many ways there are to dress it. Then ask for the list, and check it together.

  2. Ask for the rule

    When the child lists cases, ask: “In what order are you writing them?” A list with a rule is a list that can be checked.

  3. A lock with a code

    Make up a lock whose code is two digits from 1, 2 and 3, and let the child find how many codes a thief would have to try.

Opens the way to

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Give me a task on counting all the cases

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