Math Detours · Grade 6 · Around the corner · 4

Pascal's triangle

One rule, "add the two numbers above", builds a triangle that counts handshakes, paths through a city, pizza toppings and the powers of 11. Tonight you build it and go looking for the patterns.

Name ____________________Date ____________

Tonight's idea

Start with a 1. Each new row begins and ends with 1, and every number in between is the sum of the two numbers diagonally above it. The rows are numbered from 0.

1
1   1
1   2   1
1   3   3   1
1   4   6   4   1
Rows 0 to 4. The 6 in row 4 is \(3 + 3\) from the row above.

Suppose you start at the top and can only step down-left or down-right. The number in each spot is the number of different routes to reach it. There is one way to reach either edge (keep going the same way). Every inside number is a sum because the routes into it come from the two spots above.

1 1 1 1 2 1 left, rightright, left
Two routes reach the middle of row 2, so the entry is 2. Every inside number counts routes the same way.

That makes the triangle a counting machine. The entry in row \(n\), position \(k\) (counting from 0) is the number of ways to choose \(k\) things out of \(n\). Choosing which 2 of 6 people shake hands, choosing which 3 of 7 toppings go on a pizza, choosing which 3 of 6 moves in a city walk go east: all of these are entries in the triangle.

Watch one: how many handshakes among 6 people?

A handshake is a choice of 2 people out of 6. Go to row 6, position 2 (remember the first position is 0).

Row 5 is 1, 5, 10, 10, 5, 1, so row 6 is 1, 6, 15, 20, 15, 6, 1. Position 2 is 15.

Check by counting: the first person shakes 5 hands, the next shakes 4 new ones, then 3, 2, 1. \(5 + 4 + 3 + 2 + 1 = 15\). Same answer. 15 is also a triangular number; you will meet that again in question 5.

Your turn

  1. Write out rows 5, 6, 7 and 8. Keep the rows lined up so you can see the diagonals.
  2. Add up each row. What do you get for rows 0 to 8? Predict the sum of row 10 without writing it out. Why does the sum double each time? (Think about routes.)
  3. Five friends need to pick 2 of themselves for a team. How many different teams are possible? Find it in the triangle, then check by listing.
  4. A city has streets in a grid. You walk from one corner to another going 4 blocks east and 2 blocks south, never backtracking. Every route is a sequence of 6 moves, of which you choose 2 to be "south". How many routes are there?
  5. The hockey stick. Start at any 1 on an edge, run your finger down a diagonal and stop anywhere. Add those numbers. The total is the number diagonally below your stopping point, on the other side, as in the picture. Now do it with the diagonal 1, 3, 6, 10, 15. Check it in your triangle from question 1, and say what these numbers are called.
    1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1
    The hockey stick: \(1 + 4 + 10 + 20 = 35\), the number diagonally below the last one, on the other side.
  6. Work out \(11^0, 11^1, 11^2, 11^3, 11^4\). Compare with the rows of the triangle. Then compute \(11^5 = 161{,}051\) and explain what went wrong with row 5.

Stretch: the triangle inside algebra

Multiply out \((a+b)^2 = (a+b)(a+b)\). You get \(a^2 + 2ab + b^2\), and the coefficients are 1, 2, 1: row 2. Now multiply out \((a+b)^3 = (a+b)(a^2 + 2ab + b^2)\) carefully and collect like terms. Which row do you get, and why do you think the triangle shows up here?

Then, for a completely different kind of surprise: write out rows 0 to 15 and shade every odd number. Or let the computer shade the evens instead at mathdetours.com/demos/pascal and try multiples of 3 and 5 too.

Answers

  1. Row 5: 1 5 10 10 5 1. Row 6: 1 6 15 20 15 6 1. Row 7: 1 7 21 35 35 21 7 1. Row 8: 1 8 28 56 70 56 28 8 1.
  2. 1, 2, 4, 8, 16, 32, 64, 128, 256: the powers of 2, so row 10 sums to \(2^{10} = 1024\). Each number in a row feeds into two numbers below it, so the total of all routes doubles with each row. Another way: each row sums all the ways of going down \(n\) steps, and each step has 2 choices.
  3. Row 5, position 2: 10 teams. (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.)
  4. Row 6, position 2: 15 routes. (Choosing which 2 of the 6 moves are south, or equally which 4 are east: position 4 is also 15.)
  5. \(1 + 3 + 6 + 10 + 15 = 35\), which is row 7, position 3 (or 4), the same 35 the picture reaches with \(1 + 4 + 10 + 20\). The diagonal 1, 3, 6, 10, 15 is the triangular numbers, the handshake numbers from the example.
  6. 1, 11, 121, 1331, 14641: exactly rows 0 to 4. Row 5 is 1 5 10 10 5 1, but 10 is not a single digit, so the two 10s carry: \(1\,5\,10\,10\,5\,1 \to 161051\). The pattern is real (\(11 = 10 + 1\), so \(11^n\) is \((10+1)^n\), which is the stretch question in disguise), it is just hidden by carrying.

Stretch. \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), coefficients 1, 3, 3, 1: row 3. When you multiply out \((a+b)(a+b)(a+b)\), each term comes from choosing \(a\) or \(b\) from each bracket, and the coefficient of \(a^2b\) counts the ways to choose which 1 of the 3 brackets gives the \(b\): 3 ways, which is row 3, position 1. Shading the odd numbers gives the Sierpinski triangle, a fractal of triangles inside triangles, repeating at every scale.

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