An interesting solution to find the number of monomials
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Note that the generating function has to be in terms of powers of x. Example: the number of ways to select r balls from a pile of three green, three white, three blue, and three gold balls is the generating function--->(x^0+x^1+x^2+x^3)^4
Here's the problem:
4. In noncommutative algebra, the term monomial refers to any arrangement of a
sequence of variables from a set. For example, in a noncommutative algebraic
structure on a set of four variables, {x,y,z,w} , examples of monomials of length
3 are xxx,xyx,xxy,zwy,wzx,...
a) Write a generating function for the number of monomials of length, n, in a
noncommutative algebraic structure on a set of four variables.
b) Find the number of monomials of length, n, in a noncommutative
algebraic structure on a set of four variables.
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Solution Summary
This solution describes the method to find the number of ways of completing a given task using the expansion/multiplication of polynomials and collecting the coefficient to represent the number of ways of completing the task.
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Let the letters are a, b, c, d.
In making a monomial of length n, let A number of a's, B number of b's C number of c's and D number of d's are used. clearly A+B+C+D=n
the minimum number of time a particular letter is used can be 0, but the maximum number of times is n. Hence the A, B, C and D vary from 0 to ...
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