Pascal Triangle Calculator
Pascal row, C(n,k), and row sum.
What it does
Pascal triangle stacks combinations into visible rows: each number adds the pair above, and row 5 reads 1 5 10 10 5 1 at a glance. Probability, lottery odds, and algebra all drink from it.
The tool above takes row n with pick k. It returns the full row, the C(n,k) value, and the row sum.
The default row 5 with k 2 gives 10 inside a row summing to 32. The row 4 case gives 6 inside 16, and row 0 holds a lone 1.
Binomial theorem texts define these coefficients as n! over k!(n-k)!, with LibreTexts and OpenStax walking the same triangle construction. Exact integer stepping keeps rows true through 20.
Rows double in sum while entries peak mid row. Symmetric picks match, C(5,2) equals C(5,3), and probabilities normalize each row by its power of two.
How to use
- Type the row n.
- Type the pick k.
- Read the full row first.
- Read C(n,k) beside it.
- Read the row sum last.
- Stay k inside 0 to n.
Formula
- n
- Row number, 0 to 20.
- k
- Position in row, 0 to n.
- 2^n
- Row sum, powers of two.
LibreTexts mathematics carries the binomial theorem with C(n,k) coefficients matching these rows, and OpenStax algebra and trigonometry mirrors the same triangle build. Integer math multiplies then divides stepwise, staying exact through row 20. k outside 0 to n has no meaning, so the tool refuses it.
Worked example
Example: row 5, pick 2
Defaults: row 5, k = 2.
- Row: n = 5.
- Build: each entry sums the two above.
- Row: 1 5 10 10 5 1.
- Pick: k = 2 gives 10.
- Sum: 2^5 = 32.
- Tool shows 1 5 10 10 5 1, C(5, 2) = 10, Sums to 32.
Reference table
| Row | Entries | Sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 10, middle | 252 | 1024 |
Limits
- Rows 0 to 20 for exactness.
- Single picks, not distributions.
- Integers only.
- k must sit inside n.
Common mistakes
- Reading k past n.
- Counting rows from 1 not 0.
- Factorial overflow by hand.
- Mixing symmetric picks.
- Expecting probabilities unnormalized.
Try it yourself
Quick checks. Answers sit tucked under each question.
Row 4, pick 2?
Row 1 4 6 4 1, C(4,2) = 6, sums to 16.
Row 0, pick 0?
Row 1, C(0,0) = 1, sums to 1.
Row 5, pick 9?
k past n has no entry: enter k from 0 to n.
Questions
Build rows how?
Each entry adds the two above it, starting from 1 at the apex. Row n holds C(n,0) through C(n,n).
Row sums?
Row sums double each level: 1, 2, 4, 8, 16, 32. The tool reports 2^n.
Used where?
Lottery odds, coin flips, committee counts, and binomial probabilities all read C(n,k) off rows.
Big rows exact?
Yes, factorials overflow fast. This tool multiplies and divides in place to stay exact through row 20.
k bigger than n?
k past n has no entry: choose k from 0 to n.
Infinite rows?
No, rows grow exponentially. Row 20 already holds 21 numbers near 184,756.