Subtopic: multinomial theorem (formula)

The Newton's binomial formula can be generalized for the case of polynomials.

= (5)

Note: The sum is extended over all values of k₁, k₂,...,k_m that give the sum of n.

Note: P(k₁, k₂,..., k_m) = is the number all possible n-permutations with the specification (k₁, k₂,..., k_m) where letter 'x₁' is present k₁ times, letter 'x₂' can be found k₂ times and so on.

In order to make sure that the formula (5) is correct, let us remove the brackets in the expression:

(

n times

The result is the sum of terms. Each term consists of n letters (k₁ letters 'x₁', k₂ letters 'x₂',..., k_m letters 'x_m'). Such a term can be considered as a n-permutation with the specification (k₁, k₂,..., k_m). We get all possible n-permutations with this specification. They are like terms in the total sum of terms. The number of like terms with this specification is equal to P(k₁, k₂,..., k_m).

It is not difficult to see that the number of terms P(k₁, k₂,..., k_m)* in the formula (5) is equal to

Actually, the number of such terms is equal to the number of positive integer solutions of the equation:

k₁ + k₂ +... + k_m = n, where k_i ϵ {0, 1, 2,..., n}.

Each solution of this equation can be linked to a n-permutation of m types of elements with unrestricted repetitions. This n-permutation has k₁ letters 'x₁', k₂ letters 'x₂' and so on. Such a link establishes a one-to-one correspondence between the set of solutions and the set of permutations. It is so because each permutation is linked to one and only one solution of the above equation. We can conclude the number of equation solutions in question is equal to ;

H/a. Expand the expression using the multinomial formula.