Πώς να υπολογίσετε το Pascal's Triangle
Τι είναι το Pascal's Triangle;
Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. It encodes binomial coefficients, combinatorics, and the coefficients of binomial expansions. Named after Blaise Pascal (1623–1662) though known much earlier.
Οδηγός βήμα προς βήμα
- 1Row 0: 1 | Row 1: 1, 1 | Row 2: 1, 2, 1 | Row 3: 1, 3, 3, 1
- 2Entry C(n,k) = entry in row n, position k = n! / (k!(n−k)!)
- 3Binomial expansion: (a+b)^n coefficients are row n of the triangle
- 4Sum of row n = 2^n; diagonal sums give Fibonacci numbers
Worked Examples
Εισαγωγή
(x+y)^4
Αποτέλεσμα
1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴
Coefficients: Row 4 = 1,4,6,4,1
Εισαγωγή
Combinations C(5,2)
Αποτέλεσμα
10
Row 5, position 2 of Pascal's Triangle
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