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In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions. [1] [2]
The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things).
In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
By expanding the product on the left-hand side, equation follows. To prove the inclusion–exclusion principle for the cardinality of sets, sum the equation over all x in the union of A 1, ..., A n. To derive the version used in probability, take the expectation in . In general, integrate the equation with respect to μ. Always use linearity in ...
(The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.) William Betz was active in the movement to reform mathematics in the United States at that time, had written many texts on elementary mathematics topics and had "devoted his life to the improvement of ...
Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI. The multinomial coefficient (, …,) is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the ...
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In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of the polynomial's coefficients (omitting zero coefficients), and the difference ...