Pascal's Triangle
Pascal's rule is the important recurrence relation
-
(3)
which can be used to prove by mathematical induction that is a natural number for all n and k, (equivalent to the statement that k! divides the product of k consecutive integers), a fact that is not immediately obvious from formula (1).
Pascal's rule also gives rise to Pascal's triangle:
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0: 1 1: 1 1 2: 1 2 1 3: 1 3 3 1 4: 1 4 6 4 1 5: 1 5 10 10 5 1 6: 1 6 15 20 15 6 1 7: 1 7 21 35 35 21 7 1 8: 1 8 28 56 70 56 28 8 1
Row number n contains the numbers for k = 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that
- (x + y)5 = 1 x5 + 5 x4y + 10 x3y2 + 10 x2y3 + 5 x y4 + 1 y5.
The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation (3) above.
Read more about this topic: Binomial Coefficient
Famous quotes containing the word pascal:
“If Montaigne is a man in the prime of life sitting in his study on a warm morning and putting down the sum of his experience in his rich, sinewy prose, then Pascal is that same man lying awake in the small hours of the night when death seems very close and every thought is heightened by the apprehension that it may be his last.”
—Cyril Connolly (1903–1974)