About Pascal’s Triangle

Each number in Pascal’s Triangle is the sum of the two numbers directly above it. The entries in row n (0-indexed) are the binomial coefficients C(n, 0), C(n, 1), …, C(n, n) used in the binomial theorem (a + b)^n expansion.

Properties of Pascal’s Triangle

Row sums are powers of 2: the sum of all entries in row n equals 2ⁿ. Row 0 sums to 1; row 4 sums to 16; row 10 sums to 1024.
Triangular numbers: the third diagonal (1, 3, 6, 10, 15, 21, …) is the sequence of triangular numbers - the number of dots needed to form an equilateral triangle.
Fibonacci numbers in diagonals: the sums of shallow diagonals (going from right to left at 45°) produce the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …
Hockey stick identity: the sum of a diagonal run of entries equals the entry one step below the run’s end in the next diagonal. Visually it looks like a hockey stick shape in the triangle.

Binomial expansion example

Row 3 of Pascal’s Triangle is: 1, 3, 3, 1. These are exactly the coefficients in:

(a + b)³ = 1a³ + 3a²b + 3ab² + 1b³

Row 4 (1, 4, 6, 4, 1) gives the expansion of (a + b)⁴, and so on for any power.

The Sierpiński triangle connection

If you color every odd number in Pascal’s Triangle one color and every even number another, the pattern that emerges is the Sierpiński triangle - a famous fractal. This unexpected connection between a simple number table and a recursive geometric fractal is one of the most visually striking patterns in elementary mathematics.

Fibonacci numbers in Pascal's Triangle

The Fibonacci sequence is hidden in the shallow diagonal sums of Pascal's Triangle. Draw diagonal lines going from right to left at a 45° angle and sum the numbers along each diagonal:

Row 0:       1               -> 1
Row 1:      1 1              -> 1
Row 2:     1 2 1             -> 1+1 = 2
Row 3:    1 3 3 1            -> 1+2 = 3
Row 4:   1 4 6 4 1           -> 1+3+1 = 5
Row 5:  1 5 10 10 5 1        -> 1+4+3 = 8
Row 6: 1 6 15 20 15 6 1      -> 1+5+6+1 = 13

The sums 1, 1, 2, 3, 5, 8, 13, 21, … are exactly the Fibonacci sequence. This connection extends to infinite depth.

Hockey stick identity

The hockey stick identity states: the sum of a consecutive diagonal run of entries equals the entry one step below and to the right of the run's end. Visually, the selected cells form the blade of a hockey stick, and the sum is the handle end.

Example: 1 + 3 + 6 + 10 = 20. In the triangle, 1, 3, 6, 10 appear in the third diagonal; 20 appears immediately below 10 in the next diagonal. The algebraic form is: C(r,r) + C(r+1,r) + C(r+2,r) + … + C(n,r) = C(n+1, r+1).

Probability and the binomial distribution

Row n of Pascal's Triangle gives the number of ways to choose each possible outcome from n independent coin flips (or any binary trial). The probability of exactly k heads in n flips is:

P(k heads) = C(n, k) × (½)ⁿ

For 4 flips (row 4: 1, 4, 6, 4, 1), the probability of 0, 1, 2, 3, 4 heads is 1/16, 4/16, 6/16, 4/16, 1/16 respectively. The coefficients directly from Pascal's Triangle are the numerators. This is why the triangle is so central to probability theory and the binomial distribution.