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In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n =6, r =2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
(One way to prove this is by induction on k using Pascal's identity.) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials.
An archetypal double counting proof is for the well known formula for the number () of k-combinations (i.e., subsets of size k) of an n-element set: = (+) ().Here a direct bijective proof is not possible: because the right-hand side of the identity is a fraction, there is no set obviously counted by it (it even takes some thought to see that the denominator always evenly divides the numerator).
Blaise Pascal in 1654 proved Pascal's identity relating (n+1) k+1 to the sums of the p th powers of the first n positive integers for p = 0, 1, 2, ..., k. The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B 0 , B 1 , B 2 ,... which provide a uniform formula for all ...
Nicolás Balmaceda Pascal Unlike his show business-adjacent siblings, Nicolás pursued a medical career. In 2019, Pedro posted on his Instagram that his younger brother was getting his PhD in ...
Pascal's original note [1] has no proof, but there are various modern proofs of the theorem. It is sufficient to prove the theorem when the conic is a circle, because any (non-degenerate) conic can be reduced to a circle by a projective transformation. This was realised by Pascal, whose first lemma states the theorem for a circle.
Pascal’s eldest child, Natalie, resides in Canada. After her father appeared on The Golden Bachelorette , Natalie addressed rumors that they weren’t related because of Pascal’s lack of pics ...