Number of arrangements of 5 things taken out of 12 things with one particular thing included

To find the number of arrangements of 5 things taken out of 12 things in which one particular thing must be included, we can use the concept of combinations.

Step 1: Identify the number of ways to choose the particular thing

Since one particular thing must be included in the arrangement, we need to first identify the number of ways to choose that particular thing from the 12 things. In this case, we have only one particular thing, so there is only one way to choose it.

Step 2: Identify the number of ways to choose the remaining 4 things

Once the particular thing is chosen, we need to determine the number of ways to choose the remaining 4 things from the remaining 11 things. This can be calculated using combinations.

The number of ways to choose 4 things out of 11 things is given by the formula C(11, 4) = 11! / (4! * (11-4)!) = 11! / (4! * 7!) = (11 * 10 * 9 * 8) / (4 * 3 * 2 * 1) = 330.

Step 3: Calculate the total number of arrangements

Finally, we need to calculate the total number of arrangements by multiplying the number of ways to choose the particular thing (1) by the number of ways to choose the remaining 4 things (330).

Total number of arrangements = 1 * 330 = 330.

Therefore, there are 330 arrangements of 5 things taken out of 12 things in which one particular thing must be included.

Summary:
To summarize the process:
1. Identify the number of ways to choose the particular thing from the total number of things.
2. Calculate the number of ways to choose the remaining things using combinations.
3. Multiply the number of ways to choose the particular thing by the number of ways to choose the remaining things to find the total number of arrangements.