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.
Long before Pascal wrote about it in 1654, the Persian mathematician and poet Omar Khayyam described the same triangle around 1100 CE for expanding binomial powers, and the Chinese mathematician Yang Hui published and popularised it in 1261 – it is still called "Yang Hui's Triangle" in China today, honouring a mathematician who lived roughly 400 years before Pascal. The triangle is far more than a historical curiosity: geneticists use exactly its binomial coefficients when calculating the probability of different combinations of inherited traits across several genes, and each row's numbers describe the possible outcomes of any process with two options repeated multiple times, from coin flips to genetic inheritance.
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: Rows & Row Sums
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