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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.
Series 5 tests are used to determine if an article can be assigned to HD1.5 'Very Insensitive Explosive'; series 6 tests are used to determine the classification of an article within Hazard Divisions 1.1, 1.2, 1.3, or 1.4; and series 7 tests are used to determine if an article can be assigned to HD 1.6 as an article containing predominantly ...
which can be used to prove by mathematical induction that () is a natural number for all integer n ≥ 0 and all integer k, a fact that is not immediately obvious from formula (1). To the left and right of Pascal's triangle, the entries (shown as blanks) are all zero. Pascal's rule also gives rise to Pascal's triangle:
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).
1: N 4/5: S 4/5: 4/5: 4/5: 4/5: 4/5: 4/5: 4/5: 4/5: 4/5: 4/5: 4/5 Notes A blank space in the table indicates that no restrictions apply. X: Indicates that explosives of different compatibility groups may not be carried on the same transport vehicle. 1: An explosive from compatibility group L shall only be carried on the same transport vehicle ...
[1] [2] David Chapman [3] and Émile Jouguet [4] originally (c. 1900) stated the condition for an infinitesimally thin detonation. A physical interpretation of the condition is usually based on the later modelling (c. 1943) by Yakov Borisovich Zel'dovich, [5] John von Neumann, [6] and Werner Döring [7] (the so-called ZND detonation model).
The following is known about the dimension of a finite-dimensional division algebra A over a field K: dim A = 1 if K is algebraically closed, dim A = 1, 2, 4 or 8 if K is real closed, and; If K is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras over K.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...