Pascal's triangle
Pick a row n and a place k in Pascal's triangle. Read the number C(n, k), see that it is the sum of the two above it, and find that every row adds up to a power of 2: the ways of choosing, of tossing coins and of walking a grid.
Core concept
Each number in Pascal's triangle is the sum of the two above it; the k-th number of row n is C(n, k), the number of ways to choose k things from n, and row n adds up to 2ⁿ
Topic: Maths in everyday life
Write 1 at the top. Each row below starts and ends with 1, and every other number is the sum of the two just above it: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1; and so on. Number the rows from n = 0 and the places in a row from k = 0. The number at row n, place k is C(n, k), the number of ways to choose k things from n: from 6 friends you can pick 2 in C(6, 2) = 15 ways. The same numbers count the ways to get k heads from n coin tosses, and they are the coefficients of (a + b)ⁿ: (a + b)³ = a³ + 3a²b + 3ab² + b³. Each row adds up to 2ⁿ, the number of ways n coins can land.
What you will learn
- Build Pascal's triangle by adding the two numbers above.
- Read C(n, k) as the number of ways to choose k from n.
- Explain why row n adds up to 2ⁿ.
Before you start
- Know powers of 2 and how to expand (a + b)².
Materials
- Squared paper to draw the first ten rows
Equipment
- None
Procedure
- Record row 6, place 2.
- Record row 6, place 3, and row 6, place 0.
- Record row 10, place 2.
How this simulation is simplified
- Rows and places are counted from 0; C(n, k) = 0 when k is larger than n (the place is not in the row).