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Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of ...
There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by (), while the number of ways to write = + + + where every a i is a nonnegative integer is ...
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The formula can be understood as follows: p k q n−k is the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are "successes" and the remaining n − k trials result in "failure".
The net effect of these two changes is by definition, =. There are two ways to look at this thought experiment, from the perspective of the system and from the perspective of local changes. For the system we can compute the overall change in flux or species concentration by adding the two control coefficient terms, thus:
The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [2] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [2]
The sum is a sum over all partitions of p. Another exact nested sum expansion for these Stirling numbers is computed by elementary symmetric polynomials corresponding to the coefficients in x {\displaystyle x} of a product of the form ( 1 + c 1 x ) ⋯ ( 1 + c n − 1 x ) {\displaystyle (1+c_{1}x)\cdots (1+c_{n-1}x)} .