The number of ways to distribute 18 different cookies to 6 people with each receiving 3 is calculated using a series of combinations, represented by the multinomial coefficient (18 choose 3) × (15 choose 3) × (12 choose 3) × (9 choose 3) × (6 choose 3).
To determine the number of ways 18 different cookies can be given to 6 people such that each person receives exactly 3 cookies, we can treat this as a combinatoric problem of distributing items into groups. The method used to solve this problem is the multinomial coefficient. The first person can be given any 3 of the 18 cookies in 18 choose 3 ways, the second person can then receive any 3 of the remaining 15 cookies in 15 choose 3 ways, and so on until the last person receives the last 3 cookies automatically.
The multiplication of these combinations gives us the total number of distinct ways to distribute the cookies, which can be calculated as the product (18 choose 3) × (15 choose 3) × (12 choose 3) × (9 choose 3) × (6 choose 3).
To get the answer we simply evaluate this expression'
The calculation exemplifies the principle that when dividing a set of different objects into smaller distinct groups, we use the multinomial coefficient. In this case, we're dividing 18 unique cookies into 6 groups of 3. We start by choosing 3 cookies for the first person, then we have a smaller pool of cookies for the next person. This process is repeated, reducing the number of available cookies each time, until we have distributed all cookies.
The fact that the order in which we distribute cookies to different people matters is reflected in the sequential reduce-and-choose steps. This is a clear demonstration of the application of combinatorial mathematics to real-life problems like distributing items evenly.