Understanding the Problem
To determine the number of ways to select at least one item from a set of 11 different items, where you can choose up to 5 items, we will use combinatorial mathematics.
Step 1: Total Selections Possible
First, we identify the possible selections:
- You can select 1 item
- You can select 2 items
- You can select 3 items
- You can select 4 items
- You can select 5 items
Step 2: Calculate Each Selection
Now, we calculate the combinations for each selection using the combination formula "n choose k," where n is the total items (11) and k is the number of items selected.
- Selecting 1 item: C(11, 1)
- Selecting 2 items: C(11, 2)
- Selecting 3 items: C(11, 3)
- Selecting 4 items: C(11, 4)
- Selecting 5 items: C(11, 5)
Step 3: Sum of All Combinations
Now, we sum all these combinations to find the total number of ways to select at least one item:
- Total = C(11, 1) + C(11, 2) + C(11, 3) + C(11, 4) + C(11, 5)
Step 4: Calculating Each Combination
- C(11, 1) = 11
- C(11, 2) = 55
- C(11, 3) = 165
- C(11, 4) = 330
- C(11, 5) = 462
Step 5: Final Calculation
Now, we add these values:
- Total = 11 + 55 + 165 + 330 + 462 = 1023
Conclusion
Thus, the total number of ways to select at least one item from 11 different items, allowing for a maximum of 5 selections, is 1023.