Understanding the Problem
To find the number of arrangements of 10 different things taken 4 at a time, where 1 particular item must always be included, we can break this problem down into simpler steps.
Step 1: Fixing the Particular Item
- Since 1 particular item must always be included, we can consider this item as a fixed choice.
- This leaves us with the task of selecting and arranging 3 more items from the remaining 9 items (since we have 10 items in total).
Step 2: Selecting Remaining Items
- We need to choose 3 items from the remaining 9 items.
- The number of ways to choose 3 items from 9 can be calculated using the combination formula, but since we need the arrangements, we can directly calculate the arrangements.
Step 3: Calculating Arrangements
- The number of arrangements of 3 items from 9 is given by the permutation formula: P(n, r) = n! / (n-r)!
- Here, n = 9 and r = 3, so we calculate P(9, 3).
Step 4: Calculation
- P(9, 3) = 9! / (9-3)! = 9! / 6! = 9 × 8 × 7 = 504.
Step 5: Total Arrangements Including the Fixed Item
- Since the fixed item is included, the total arrangements of 4 items (1 fixed + 3 selected) is simply P(9, 3).
- Therefore, the total number of arrangements is 504.
Final Result
- The number of arrangements of 10 different things taken 4 at a time, with 1 particular item always included, is 504.