The number of arrangements of 5 things taken out of 12 things in which one particular thing must always be included

To solve this problem, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order. In this case, we need to find the number of arrangements of 5 things taken out of 12 things, where one particular thing must always be included.

Step 1: Identify the given values
- Total number of things (n) = 12
- Number of things to be selected (r) = 5
- One particular thing that must always be included = 1

Step 2: Calculate the number of arrangements
To find the number of arrangements, we can use the formula for permutations:

P(n, r) = n! / (n - r)!

In this case, we need to consider the one particular thing as already included, so we have 11 remaining things to choose from.

P(11, 4) = 11! / (11 - 4)!

Simplifying this expression:

P(11, 4) = 11! / 7!

Step 3: Calculate the factorial
To further simplify the expression, we need to calculate the factorial of 11 and 7.

11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1

Step 4: Calculate the number of arrangements
Now, we can substitute the factorial values into the formula:

P(11, 4) = (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (7 x 6 x 5 x 4 x 3 x 2 x 1)

By canceling out common terms:

P(11, 4) = 11 x 10 x 9 x 8

Step 5: Calculate the final result
Calculating the expression:

P(11, 4) = 11 x 10 x 9 x 8
P(11, 4) = 7,920

Therefore, the number of arrangements of 5 things taken out of 12 things, where one particular thing must always be included, is 7,920.

Summary:
The number of arrangements of 5 things taken out of 12 things, where one particular thing must always be included, is 7,920. This is calculated using the formula for permutations and considering the remaining 11 things after including the particular thing. The factorial of 11 and 7 is calculated to simplify the expression, resulting in the final answer of 7,920.