Concepts · 3 min read
Pascal's Triangle and the Binomial Theorem Explained
Building the triangle
| Row n | Entries | Sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 2 | 1 2 1 | 4 |
| 3 | 1 3 3 1 | 8 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 6 | 1 6 15 20 15 6 1 | 64 |
| 7 | 1 7 21 35 35 21 7 1 | 128 |
| 8 | 1 8 28 56 70 56 28 8 1 | 256 |
Every row sums to a power of 2: row n sums to 2ⁿ.
Entries are combinations
The k-th entry of row n is nCr: row 5, entry 2 is C(5,2) = 10. Compute any with the permutation and combination calculator.
5C2
- 5C2 = 10
Step-by-step working 3 steps
- 1Order does not matter, no repeats
nCr = n! / (r!(n−r)!) = nPr / r!
Use combinations when you only care which items are chosen (committees, lottery picks).
- 2Compute nPr and r!
5P2 = 20 2! = 2
- 3Divide
20 ÷ 2 = 10
The binomial theorem
(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ. So (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴, using row 4: 1, 4, 6, 4, 1.
Numerical check with a = 1, b = 2: (1 + 2)⁴ = 81, and 1×1 + 4×2 + 6×4 + 4×8 + 1×16 = 81.
Why it matters
The same numbers count paths, subsets and coin-flip outcomes — see lottery odds and combinations.
Frequently asked questions
How do I find the 10th row quickly?
Use nCr: entry k of row 10 is C(10, k).
Why does each row sum to 2ⁿ?
Setting a = b = 1 in (a + b)ⁿ gives 2ⁿ = ΣC(n,k).
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