91Ó°ÊÓ

The multinomial expansion is an extension of the binomial theorem, which deals with the power of sums of more than two terms. In this example, we are expanding \((x+y+z)^{10}\), which is a sum of three variables raised to the 10th power. This process allows us to write the expression as a sum of many different terms.

The multinomial theorem states that any term in the expansion of \((x_1 + x_2 + ... + x_m)^n \) takes the form:
\[ \frac{n!}{k_1!k_2!...k_m!} x_1^{k_1} x_2^{k_2} ... x_m^{k_m} \]
where \(k_1 + k_2 + ... + k_m = n\).

To find the term we are interested in, we locate the exponents of \(x, y,\) and \(z\) that sum up to 10 (since we are expanding to the 10th power). For the term \(x^3 y^2 z^5\), the exponents are \(a=3\), \(b=2\), and \(c=5\).

Factorials are a crucial concept when working with the multinomial theorem and combinatorial mathematics in general. A factorial is denoted by an exclamation mark \(!\) and represents the product of all positive integers up to that number.

For example:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
In this exercise, we need to calculate the following factorials:

• \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(2! = 2 \times 1 = 2\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

These factorials are used to determine the coefficient of the term we are looking for by substituting into the formula.

Combinatorial coefficients, also known as multinomial coefficients, are the factors that appear in the expansion. They represent the number of ways to arrange the variables in a given term.

For our specific term \(x^3 y^2 z^5\) in the expansion of \((x+y+z)^{10}\), the coefficient is calculated using the formula:
\[ \frac{10!}{3!2!5!} \]
After calculating the factorials in the previous section, we substitute them into the formula:
\[ \frac{3628800}{6 \times 2 \times 120} = \frac{3628800}{1440} = 2520 \]
This shows that the coefficient for the term \(x^3 y^2 z^5\) in the expansion of \((x+y+z)^{10}\) is 2520.