Building the triangle
Pascal’s triangle starts with a single 1 at the top. Every later row starts and ends with 1, and each number in between is the sum of the two numbers directly above it. The first rows look like this, counting the top row as row 0:
- 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
Check one entry: in row 5, the 10 is the sum of the 4 and 6 above it. In row 6, the 20 is 10 + 10.
The triangle is named after Blaise Pascal, who wrote about it in the seventeenth century, but the pattern was known earlier in other parts of the world.
What the numbers count
The entry in row n, position k (counting positions from 0) is the number of ways to choose k items from n items, written C(n, k) and read “n choose k”. These are the binomial coefficients.
A concrete check: row 4 reads 1, 4, 6, 4, 1. Choosing 2 players from a team of 4 can be done in 4 × 3 ÷ 2 = 6 ways, and 6 is the middle entry of row 4. Choosing 1 player gives 4 ways, and choosing none or all four gives 1 way each. The formula behind this is the one used for combinations and permutations in general, where order is divided out.
Why the adding rule works
Pick one particular item, call it Z, from a set of n items. Any group of k items either contains Z or does not.
- Groups that contain Z: choose the other k − 1 from the remaining n − 1 items, C(n − 1, k − 1) ways.
- Groups that do not contain Z: choose all k from the remaining n − 1 items, C(n − 1, k) ways.
Adding the two cases gives C(n, k) = C(n − 1, k − 1) + C(n − 1, k). That is exactly the rule “each number is the sum of the two above it”. The triangle is a table of counts that happens to build itself.
A few other patterns are worth knowing:
- Each row is symmetric, because choosing k items to include is the same as choosing n − k to leave out.
- Row n adds up to 2ⁿ. Row 4: 1 + 4 + 6 + 4 + 1 = 16 = 2⁴. Row 5: 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2⁵.
From counts to coin toss probabilities
Toss a fair coin 4 times. There are 2 × 2 × 2 × 2 = 16 equally likely sequences of heads and tails. How many contain exactly k heads? That is choosing which k of the 4 tosses are heads, so the answer is C(4, k): the entries of row 4.
| Heads | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Sequences | 1 | 4 | 6 | 4 | 1 |
| Probability | 1/16 | 4/16 | 6/16 | 4/16 | 1/16 |
So the chance of exactly 2 heads in 4 tosses is 6/16 = 3/8 = 0.375, or 37.5 percent. The chance of exactly 4 heads is 1/16 = 6.25 percent. These are illustrations for a fair coin whose tosses are independent.
For 5 tosses, row 5 gives 1, 5, 10, 10, 5, 1 out of 32 sequences. The chance of at least 4 heads is (5 + 1) ÷ 32 = 6/32 = 3/16 = 18.75 percent.
The symmetry of the row mirrors the symmetry of a fair coin: 0 heads is as likely as 5 heads, and 2 as likely as 3.
Connecting to the binomial distribution
Reading a row as probabilities (dividing by 2ⁿ) is the simplest version of the binomial distribution. When the coin is biased, or when you count successes in repeated independent trials such as dice showing a particular face, the row of the triangle still counts the arrangements. Each arrangement just has a different probability, found by multiplying the chance of each trial. The triangle supplies the “how many ways” part; the probabilities supply the “how likely each way is” part.
The shape of the rows also hints at why bell curves arise. The middle entries are the biggest and the ends the smallest, which is the start of the pattern shown by the normal distribution when n gets large.
Why the name “binomial coefficient”
When you expand (a + b)ⁿ, the coefficients of the terms are the entries of row n. For example, (a + b)² = a² + 2ab + b², matching row 2: 1, 2, 1. And (a + b)³ = a³ + 3a²b + 3ab² + b³, matching row 3. Each term corresponds to choosing which of the n brackets contribute an a and which a b, which is the same choice as picking heads and tails positions.
A caution about streaks
The triangle describes counts for a fixed number of tosses. It does not say that after several heads a tail is due. Each toss of a fair coin is independent, and a run of heads is just one of the 2ⁿ sequences, equally likely as any other specific sequence. Mixing these up is the gambler’s fallacy. For a closer look at runs in a series of tosses, see coin toss streaks. Games of chance with a house edge cost players money on average whatever pattern appears, and gambling can be harmful. A local support service can help anyone struggling with it.