Multinomial Theorem on Probability

In probability theory, the multinomial theorem is a generalization of the binomial theorem to multinomial coefficients. It provides a way to expand a multinomial expression raised to a positive integer power. The multinomial theorem is a powerful tool in probability calculations, especially when dealing with experiments that have multiple outcomes.

Multinomial Coefficients

Before we dive into the multinomial theorem, let's first understand multinomial coefficients. In combinatorics, the multinomial coefficient is a way to count the number of ways to partition a set of objects into groups of different sizes. The multinomial coefficient is denoted by $\binom{n}{k_1, k_2, ..., k_r}$, where $n$ is the total number of objects and $k_1, k_2, ..., k_r$ are the sizes of the groups.

The formula for the multinomial coefficient is given by:

$$\binom{n}{k_1, k_2, ..., k_r} = \frac{n!}{k_1! \cdot k_2! \cdot ... \cdot k_r!}$$

where $n!$ denotes the factorial of $n$.

Multinomial Theorem

The multinomial theorem provides a way to expand a multinomial expression raised to a positive integer power. It states that:

$$(x_1 + x_2 + ... + x_r)^n = \sum_{k_1 + k_2 + ... + k_r = n} \binom{n}{k_1, k_2, ..., k_r} \cdot x_1^{k_1} \cdot x_2^{k_2} \cdot ... \cdot x_r^{k_r}$$

where $x_1, x_2, ..., x_r$ are variables, $n$ is a positive integer, and the summation is taken over all possible values of $k_1, k_2, ..., k_r$ that satisfy the condition $k_1 + k_2 + ... + k_r = n$.

Differences between Binomial Theorem and Multinomial Theorem

The multinomial theorem is a generalization of the binomial theorem. While the binomial theorem deals with expressions of the form $(a + b)^n$, the multinomial theorem extends this concept to expressions of the form $(x_1 + x_2 + ... + x_r)^n$. Here are some key differences between the two theorems:

Binomial Theorem	Multinomial Theorem
Deals with two variables ($a$ and $b$)	Deals with multiple variables ($x_1, x_2, ..., x_r$)
Expansion is in terms of binomial coefficients	Expansion is in terms of multinomial coefficients
Expression is raised to a positive integer power	Expression is raised to a positive integer power
Number of terms in the expansion is $n+1$	Number of terms in the expansion is $\binom{n+r-1}{r-1}$

Example

Let's consider an example to understand the multinomial theorem better. Suppose we have a bag containing 5 red balls, 3 blue balls, and 2 green balls. We want to find the number of ways to choose 4 balls from the bag.

Using the multinomial theorem, we can express this as:

$$(x_1 + x_2 + x_3)^4$$

where $x_1$ represents the red balls, $x_2$ represents the blue balls, and $x_3$ represents the green balls.

Expanding this expression using the multinomial theorem, we get:

$$(x_1 + x_2 + x_3)^4 = \binom{4}{0, 0, 4} \cdot x_1^0 \cdot x_2^0 \cdot x_3^4 + \binom{4}{0, 1, 3} \cdot x_1^0 \cdot x_2^1 \cdot x_3^3 + \binom{4}{0, 2, 2} \cdot x_1^0 \cdot x_2^2 \cdot x_3^2 + \binom{4}{1, 0, 3} \cdot x_1^1 \cdot x_2^0 \cdot x_3^3 + \binom{4}{1, 1, 2} \cdot x_1^1 \cdot x_2^1 \cdot x_3^2 + \binom{4}{1, 2, 1} \cdot x_1^1 \cdot x_2^2 \cdot x_3^1 + \binom{4}{2, 0, 2} \cdot x_1^2 \cdot x_2^0 \cdot x_3^2 + \binom{4}{2, 1, 1} \cdot x_1^2 \cdot x_2^1 \cdot x_3^1 + \binom{4}{3, 0, 1} \cdot x_1^3 \cdot x_2^0 \cdot x_3^1 + \binom{4}{4, 0, 0} \cdot x_1^4 \cdot x_2^0 \cdot x_3^0$$

Simplifying this expression, we get:

$$x_3^4 + 4x_2x_3^3 + 6x_2^2x_3^2 + 4x_1x_3^3 + 12x_1x_2x_3^2 + 6x_1^2x_2x_3 + 4x_1^2x_3^2 + 12x_1^2x_2x_3 + 6x_1^3x_2 + x_1^4$$

This expansion gives us the number of ways to choose 4 balls from the bag, taking into account the different colors of the balls.

Conclusion

The multinomial theorem is a powerful tool in probability calculations, especially when dealing with experiments that have multiple outcomes. It allows us to expand multinomial expressions raised to a positive integer power, providing a way to count the number of ways to partition a set of objects into groups of different sizes. By understanding the multinomial theorem and its application, we can solve complex probability problems more efficiently.