Binomial Theorem

The Binomial Theorem

The binomial theorem gives a formula for expanding expressions of the form (a + b)ⁿ:

(a + b)ⁿ = Σ(k=0 to n) C(n,k) × aⁿ⁻ᵏ × bᵏ

Example: (x + y)³
= C(3,0)x³ + C(3,1)x²y + C(3,2)xy² + C(3,3)y³
= x³ + 3x²y + 3xy² + y³

Pascal's Triangle

Pascal's triangle is a geometric arrangement of binomial coefficients. 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

Properties:
  Row n has n + 1 entries
  Entry k in row n = C(n, k)
  Sum of row n = 2ⁿ
  Symmetric: C(n,k) = C(n, n-k)

Applications

1. Finding specific coefficients:
   Coefficient of x² in (1 + x)⁵ = C(5, 2) = 10

2. Counting subsets:
   Total subsets of an n-element set = Σ C(n,k) = 2ⁿ

3. Number of paths in a grid:
   Paths from (0,0) to (m,n) = C(m+n, m)

4. Sum of binomial coefficients:
   C(n,0) + C(n,1) + ... + C(n,n) = 2ⁿ