Pascal's Triangle Calculator
Visualize the famous mathematical triangle or calculate specific entries (nCr). This tool is perfect for algebra expansions and probability study.
Row Sum: Σ C(n, k) = 2ⁿ
Entry Formula: C(n, k) = n! / [k! × (n-k)!]
Row 4: 1, 4, 6, 4, 1 Sum: 1+4+6+4+1 = 16 (which is 2⁴)
What is Pascal's Triangle?
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. The edges are always 1.
What do the numbers in Pascal's Triangle represent?
The numbers are the binomial coefficients (n choose k). For example, the 4th row (1, 4, 6, 4, 1) contains the coefficients for expanding (x + y)⁴.
How do I find a specific entry in the triangle?
The value at row n and position k (starting from 0) is calculated using the combination formula: C(n, k) = n! / [k! * (n - k)!].
What are some patterns in Pascal's Triangle?
Patterns include: 1) The sum of each row n is 2ⁿ. 2) The diagonals contain counting numbers, triangular numbers, and tetrahedral numbers. 3) It is symmetrical.
🔗 Related Calculators
📐 Formula
Row Sum: Σ C(n, k) = 2ⁿ
Entry Formula: C(n, k) = n! / [k! × (n-k)!]
📝 Example Calculation
Row 4: 1, 4, 6, 4, 1 Sum: 1+4+6+4+1 = 16 (which is 2⁴)
❓ Frequently Asked Questions
What is Pascal's Triangle?▼
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. The edges are always 1.
What do the numbers in Pascal's Triangle represent?▼
The numbers are the binomial coefficients (n choose k). For example, the 4th row (1, 4, 6, 4, 1) contains the coefficients for expanding (x + y)⁴.
How do I find a specific entry in the triangle?▼
The value at row n and position k (starting from 0) is calculated using the combination formula: C(n, k) = n! / [k! * (n - k)!].
What are some patterns in Pascal's Triangle?▼
Patterns include: 1) The sum of each row n is 2ⁿ. 2) The diagonals contain counting numbers, triangular numbers, and tetrahedral numbers. 3) It is symmetrical.