Pascal's Triangle

What is Pascal's Triangle?

Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.

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

Rule: Each entry = sum of the two entries above it.

Example: The 6 in Row 4 = 3 + 3 (from Row 3)

Building the Triangle

Step 1: Start with 1 at the top (Row 0).

Step 2: Each row starts and ends with 1.

Step 3: Every interior number is the sum of the two numbers above it.

Row 3:   1   3   3   1
           ↘ ↙ ↘ ↙
Row 4:   1   4   6   4   1

• 4 = 1 + 3
• 6 = 3 + 3
• 4 = 3 + 1

Connection to Combinations: C(n, r)

The entry in row n at position r equals C(n, r):

C(n, r) = n! / [r! × (n−r)!]

Row 4:  C(4,0)  C(4,1)  C(4,2)  C(4,3)  C(4,4)
          1       4       6       4       1

C(4, 2) = 4! / (2! × 2!) = 24 / 4 = 6 ✓

This means: Pascal's triangle lists all combination values!

Key Patterns

1. Row Sum = Powers of 2

Each row sums to 2ⁿ (where n = row number):

Row 0: 1              = 2⁰ = 1
Row 1: 1+1            = 2¹ = 2
Row 2: 1+2+1          = 2² = 4
Row 3: 1+3+3+1        = 2³ = 8
Row 4: 1+4+6+4+1      = 2⁴ = 16

2. Symmetry

Each row is symmetric: C(n, r) = C(n, n−r)

Example: C(5, 2) = C(5, 3) = 10

3. Triangular Numbers (Column 2)

The third diagonal: 1, 3, 6, 10, 15, 21, ... (triangular numbers)

4. Fibonacci Numbers

Sum along the diagonals gives the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ...

Binomial Theorem Connection

Pascal's Triangle gives the coefficients of (a + b)ⁿ:

(a + b)⁰ = 1

(a + b)¹ = 1a + 1b

(a + b)² = 1a² + 2ab + 1b²

(a + b)³ = 1a³ + 3a²b + 3ab² + 1b³

(a + b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴

The coefficients 1, 4, 6, 4, 1 are Row 4 of Pascal's Triangle!

Using C(n, r) = C(n−1, r−1) + C(n−1, r)

The triangle rule in formula form:

Each entry = entry above-left + entry above-right

Example: Find C(6, 3) using the triangle rule.

C(6, 3) = C(5, 2) + C(5, 3)
        = 10 + 10
        = 20 ✓

Applications

Probability

Probability of exactly k heads in n coin flips:

P(k heads) = C(n, k) / 2ⁿ

Example: Probability of exactly 2 heads in 4 flips:

P = C(4,2) / 2⁴ = 6/16 = 3/8

Counting Paths on a Grid

The number of shortest paths from the top-left to each cell in a grid follows Pascal's Triangle.

Practice