Pascal's Triangle - A Triangle Full of Patterns
Pascal's Triangle is a triangular arrangement of numbers in which each number is the sum of the two numbers directly above it. Named after the French mathematician Blaise Pascal (though known centuries earlier in China, India, and Persia), it contains an extraordinary number of hidden patterns and connects to probability, algebra, and combinatorics.
Building Pascal's Triangle
Start with 1 at the top. Each row begins and ends with 1. Every other number equals the sum of the two numbers above it.
| Row | Numbers |
|---|---|
| Row 0 | 1 |
| Row 1 | 1 1 |
| Row 2 | 1 2 1 |
| Row 3 | 1 3 3 1 |
| Row 4 | 1 4 6 4 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 |
Hidden Patterns
| Pattern | Where to find it |
|---|---|
| Natural numbers | Second diagonal from the left: 1, 2, 3, 4, 5 … |
| Triangular numbers | Third diagonal: 1, 3, 6, 10, 15 … |
| Powers of 2 | Row sums: 1, 2, 4, 8, 16, 32 … (each row sum = 2row number) |
| Powers of 11 | Rows 0–4 read as single numbers: 1, 11, 121, 1331, 14641 |
| Fibonacci numbers | Diagonal sums of Pascal's Triangle produce the Fibonacci sequence |
| Binomial coefficients | Row n gives the coefficients of (a+b)n |
Binomial Expansion
Row n of Pascal's Triangle gives the coefficients when expanding (a + b)n.
Row 4: 1, 4, 6, 4, 1.
(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4.
Pascal's Triangle and Probability
Row n of Pascal's Triangle counts the number of ways to get each outcome when tossing a coin n times. For 3 tosses (Row 3: 1, 3, 3, 1): there is 1 way to get 3 heads, 3 ways to get 2 heads, 3 ways to get 1 head, and 1 way to get 0 heads. Total outcomes = 1+3+3+1 = 8 = 23.
Worked Examples
Row 7: 1, 7, 21, 35, 35, 21, 7, 1.
Row 8: 1, (1+7), (7+21), (21+35), (35+35), (35+21), (21+7), (7+1), 1
= 1, 8, 28, 56, 70, 56, 28, 8, 1.
Sum check: 1+8+28+56+70+56+28+8+1 = 256 = 28 ✓
Row 5: 1, 5, 10, 10, 5, 1.
(x + 1)5 = x5 + 5x4 + 10x3 + 10x2 + 5x + 1.
Key Takeaways
- Each entry = sum of the two entries directly above it. Edges are always 1.
- Row n sums to 2n.
- Row n gives binomial coefficients for (a+b)n.
- The triangle contains natural numbers, triangular numbers, Fibonacci numbers, and powers of 11 – all hidden inside.
Practice Questions
- Write Rows 8 and 9 of Pascal's Triangle.
- Verify that the sum of Row 9 equals 29.
- Use Pascal's Triangle to expand (a + b)6.
- Identify the triangular numbers in the first eight rows of Pascal's Triangle.
- How many ways can you get exactly 2 heads when tossing a fair coin 5 times? Use Pascal's Triangle to answer.
You Have Completed the Patterns and Sequences Section!
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