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Maths olympiad lesson: counting grid routes with Pascal's triangle

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

Grid routes: practice

Routes go along grid lines, only right or up, from (0, 0). Show your working on paper. Questions 6 and 9 need a written explanation.

Diagnostic question: choose and explain

  1. How many shortest routes are there from (0, 0) to (5, 2)?
    1. 21
    2. 42
    3. 128
    4. 10

Practice questions

  1. How many shortest routes are there from (0, 0) to (2, 2)?
  2. Row 6 of Pascal's triangle: 1, 6, ___, ___, 15, 6, 1
  3. How many shortest routes are there from (0, 0) to (5, 3)?
  4. How many different 7-letter words can be made from four A's and three B's (such as AABABBA)?
  5. How many shortest routes from (0, 0) to (5, 3) pass through (2, 2)?
  6. Explain why the number of routes to a corner is the number of routes to the corner on its left plus the number to the corner below it. Why is no route counted twice, and none missed?
  7. How many shortest routes from (0, 0) to (4, 4) avoid the corner (2, 2)?
  8. How many shortest routes from (0, 0) to (4, 4) never go above the line y = x? (They may touch it.)
  9. Stretch: every route from (0, 0) to (3, 3) crosses the line x + y = 3 at exactly one corner. Use this to explain why 1² + 3² + 3² + 1² = 20. Then say what 1² + 4² + 6² + 4² + 1² must be, and why.

Support: For question 1: draw the 2 by 2 grid, put 1 on every corner of the bottom edge and the left side, then add. For the olympiad questions: make the grid smaller, count by adding, and look for the pattern before you prove it.

Grid routes: exit

A number and two sentences.

  1. How many shortest routes from (0, 0) to (4, 3) avoid the corner (1, 1)? Explain in two sentences why you have counted every route exactly once.

Exit check

One count with a written reason shows who can count without missing or repeating.

Routes from (0, 0) to (4, 3) that avoid the corner (1, 1).

Grid routes: count and explain

Routes go along grid lines, only right or up. Count each one two ways where you can (adding on a grid, and choosing). Question 6 is a written explanation: write it as you would in an olympiad.

1. How many shortest routes are there from (0, 0) to (3, 4)?

2. Write out row 7 of Pascal's triangle. What do its numbers add up to, and why?

3. How many shortest routes from (0, 0) to (6, 2) pass through (3, 1)?

4. How many shortest routes from (0, 0) to (5, 5) avoid the corner (2, 2)?

5. In 3D, a route moves one unit at a time in the x, y or z direction, never backwards. How many shortest routes go from (0, 0, 0) to (2, 1, 1)?

6. Explain why there are as many routes to (a, b) as to (b, a), without working either number out.

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