MULTINOMIAL

Application:

The multinomial coefficient is a powerful tool for calculating the number of ways to arrange a set of objects into distinct groups. It's an extension of the binomial coefficient, which deals with two groups. The formula for the multinomial coefficient is:

$\left(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m}\right)=\frac{n!}{k_1!k_2!\cdots k_m!}$

where $n$ is the total number of items, and $k_1,k_2,\cdots ,k_m$ are the sizes of the distinct groups, such that $k_1+k_2+\cdots +k_m=n$.

Let's consider a scenario: a company is forming a new product development team.

The company has a pool of 10 employees and needs to form a team of 10 with the following roles:

• 3 software engineers
• 4 data scientists
• 2 UX designers
• 1 project manager

The question is, how many different ways can the company assign these 10 employees to these specific roles?

Here's how we can use the MULTINOMIAL function:

The MULTINOMIAL function takes the sizes of the distinct groups as its arguments. The total number of items (n) is calculated automatically by the function.

• Group 1: 3 software engineers
• Group 2: 4 data scientists
• Group 3: 2 UX designers
• Group 4: 1 project manager

The sum of the group sizes is $3+4+2+1=10$, which equals the total number of employees.

Using the MULTINOMIAL function, the calculation is:

$\text{MULTINOMIAL}(3,4,2,1)=\frac{10!}{3!\times 4!\times 2!\times 1!}$

Let's break down the factorials:

• $10!=3,628,800$
• $3!=3\times 2\times 1=6$
• $4!=4\times 3\times 2\times 1=24$
• $2!=2\times 1=2$
• $1!=1$

Now, substitute these values back into the formula:

$\frac{3,628,800}{6\times 24\times 2\times 1}=\frac{3,628,800}{288}=12,600$

This means there are 12,600 different ways to form the new product development team with the specified roles.

Summary Table: