Question

If there are 6 alike fruits, 7 alike vegetables, and 8 alike biscuits, then the number of ways of selecting any number of things out of them such that at least one from each category is selected, is:

• A. \( 504 \)
• B. \( 336 \)
• C. \( 503 \)
• D. \( 335 \)

Hint:

When selecting from alike objects, the number of ways to pick at least one from a set of \( n \) identical items is simply \( n \), since we can choose any number from 1 to \( n \).

Answer 1

The Correct Option is B

Step 1: Understanding the selection process
We have three categories of items: - 6 alike fruits, - 7 alike vegetables, - 8 alike biscuits. We need to determine the number of ways to select at least one item from each category. Step 2: Finding possible selections from each category
Since the items in each category are identical, selecting any number from each category corresponds to choosing a subset. For each category: - The number of ways to select at least one fruit: \( 6 \) (choose from 1 to 6). - The number of ways to select at least one vegetable: \( 7 \) (choose from 1 to 7). - The number of ways to select at least one biscuit: \( 8 \) (choose from 1 to 8). Step 3: Applying the multiplication principle
Since the selections from each category are independent, the total number of ways is given by: \[ 6 \times 7 \times 8 = 336. \] Step 4: Conclusion
Thus, the total number of ways to make a selection while ensuring at least one item from each category is: \[ \boxed{336}. \]