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This essential trains for: AMC-10, AMC-12, AIME.

Consider the expression:

equation

and derive a formula for expanding it into a sum of monomial terms.

First of all, each term is going to be of degree n, i.e. the exponents of the three variables must add up to n:

equation

Each variable's exponent will vary, subject to the above condition, from 0 to n.

The coefficient of the generic term:

equation

can be calculated by counting in how many ways one can obtain such a product.

The number of ways is given by the multiplications that, due to the commutative property, result in the same exponents. From among a x-s, b y-s and c z-s we can form κ(a,b,c) distinct strings:

equation

equation

Since:

equation

we have:

equation

and the formula becomes:

equation

Example: What are the coefficients of x55 and x54 after expanding and collecting terms in the expression:

equation

Solution:

We notice that there is only one way of producing an exponent of 54:

equation

and there is no way of producing an exponent of 55. Therefore, the coefficient of x55 is zero.

The coefficient of x55 is given by:

equation