Category:
Math
All
BiologyChemistryConstructionConversionEcologyEveryday LifeFinanceFoodHealthPhysicsSportsStatisticsOther

Pascal's Triangle Calculator

Generate Pascal's triangle, calculate binomial coefficients, and explore mathematical patterns

Calculate Pascal's Triangle

Row index (0-based, max 20 for display)

Number of triangle rows to show

Pascal's Triangle

                      1
                    1   1
                  1   2   1
                1   3   3   1
              1   4   6   4   1
            1   5  10  10   5   1
          1   6  15  20  15   6   1
        1   7  21  35  35  21   7   1
      1   8  28  56  70  56  28   8   1
    1   9  36  84 126 126  84  36   9   1
  1  10  45 120 210 252 210 120  45  10   1
[1, 5, 10, 10, 5, 1]
Row 5
32
Row Sum (2^5)
6
Entries in Row

Formula: C(n,k) = n! / (k! × (n-k)!) - combinations of k items from n items

Pattern: Each entry = sum of two entries above it in previous row

Applications: Combinatorics, probability, binomial expansion, algebra

Pattern Analysis for Row 5

Mathematical Properties

Symmetry:Yes
Row sum:32
Max value:10
Max position:2
Entries count:6

Binomial Expansion

(x + y)^5 =
x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5
The coefficients match row 5 of Pascal's triangle

Interesting Facts

  • • Sum of row n equals 2^n = 32
  • • Each row is symmetric around the center
  • • Maximum value occurs at the center (or centers for even rows)
  • • Row 5 has 6 entries
  • • Each entry equals C(n,k) = number of ways to choose k items from n

Famous Pascal Triangle Rows

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]

Pascal's Triangle Properties

•

Each number is the sum of the two numbers above it

•

Every row is symmetric

•

Sum of row n equals 2ⁿ

•

Entries are binomial coefficients C(n,k)

•

Coefficients for (x+y)ⁿ expansion

Applications

Combinatorics
Count combinations and arrangements
Probability
Calculate probabilities in binomial distributions
Algebra
Expand binomial expressions (x+y)ⁿ
Number Theory
Study patterns and mathematical sequences

Understanding Pascal's Triangle

What is Pascal's Triangle?

Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. Named after French mathematician Blaise Pascal, it has profound connections to combinatorics, probability, and algebra.

How to Build It

  1. Start with 1 at the top (row 0)
  2. Each row starts and ends with 1
  3. Every other number is the sum of the two numbers above it
  4. Continue this pattern to build more rows

Construction Rule

C(n,k) = C(n-1,k-1) + C(n-1,k)

Key Applications

  • 🎲Probability: Calculate odds in coin flips, dice games
  • 🧮Combinatorics: Count ways to choose items from sets
  • 📐Algebra: Expand expressions like (x+y)ⁿ
  • 🔢Number Theory: Study Fibonacci numbers, powers of 2
  • 💻Computer Science: Algorithms, data structures

Real-World Example

Problem: How many ways can you choose 3 movies from 20 favorites?

Solution: Use C(20,3) = 1,140 different combinations

Related Math Calculators

Fibonacci Calculator

Generate Fibonacci sequences and explore golden ratio relationships

Arithmetic Sequence Calculator

Calculate arithmetic progressions and their sums

Sum of Series Calculator

Calculate various mathematical series and their convergence