There are three types of meals: chicken, beef, and lobster. If eight people eat dinner together, in how many different ways can 3 order chicken, 4 order beef, and 1 order lobster?

This is a permutation problem involving multiple groups. The formula for this is:

P(n; n1, n2, ..., nk) = n! / (n1! * n2! * ... * nk!)

where:

• n is the total number of items
• n1, n2, ..., nk are the numbers of each type of item

In this case:

• n = 8 (the total number of people)
• n1 = 3 (the number of people ordering chicken)
• n2 = 4 (the number of people ordering beef)
• n3 = 1 (the number of people ordering lobster)

Substituting these values into the formula, we get:

P(8; 3, 4, 1) = 8! / (3! * 4! * 1!)

Calculating the above expression gives 560.

So, there are 560 different ways for the eight people to order their meals.

This is a permutation problem involving multiple groups. The formula for this is:

P(n; n1, n2, ..., nk) = n! / (n1! * n2! * ... * nk!)

where:

• n is the total number of items
• n1, n2, ..., nk are the numbers of each type of item

In this case:

• n = 8 (the total number of people)
• n1 = 3 (the number of people ordering chicken)
• n2 = 4 (the number of people ordering beef)
• n3 = 1 (the number of people ordering lobster)

So, we can plug these values into the formula:

P(8; 3, 4, 1) = 8! / (3! * 4! * 1!)

First, calculate 8!, which is 8765432*1 = 40320.

Next, calculate 3!, which is 321 = 6.

Then, calculate 4!, which is 432*1 = 24.

Finally, 1! is just 1.

So, the number of different ways the meals can be ordered is:

P(8; 3, 4, 1) = 40320 / (6 * 24 * 1) = 280.

To solve this problem, we need to use the multinomial coefficient, which is a generalization of the binomial coefficient. It is used when there are more than two categories to choose from, and we want to know how many ways we can choose a certain number from each category.

The formula for the multinomial coefficient is:

C(n; k1, k2, ..., km) = n! / [k1! * k2! * ... * km!]

Where:

• n is the total number of items (in this case, 8 people)
• k1, k2, ..., km are the numbers of each type of item to choose (in this case, 3 chicken, 4 beef, and 1 lobster)
• "!" denotes a factorial, meaning the product of all positive integers up to that number.

Substituting these into the formula gives:

C(8; 3, 4, 1) = 8! / [3! * 4! * 1!] = (876) / (321) * (432*1) * 1 = 560

So there are 560 different ways for the eight people to order their meals.