Question

There are \(4\) oranges, \(5\) apples, \(7\) mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

• A. \(210\)
• B. \(240\)
• C. \(209\)
• D. \(239\)

Hint:

If an item can be selected from \(0\) to \(n\) times, then the number of choices is \[ n+1. \] When the question asks for “at least one”, subtract the empty selection at the end.

Answer 1

The Correct Option is D

Step 1: Understand the selection type.
There are 4 identical oranges, 5 identical apples, 7 identical mangoes. When fruits of the same type are identical, the number of ways to pick oranges is: pick 0, 1, 2, 3, or 4 oranges = 5 choices.

Step 2: Count orange selections.
4 oranges $\Rightarrow$ 5 choices (0 through 4).

Step 3: Count apple selections.
5 apples $\Rightarrow$ 6 choices (0 through 5).

Step 4: Count mango selections.
7 mangoes $\Rightarrow$ 8 choices (0 through 7).

Step 5: Apply multiplication principle, then subtract the empty case.
Total selections including choosing nothing: $5 \times 6 \times 8 = 240$. Subtract the one case where no fruit is chosen: $240 - 1 = 239$.

Step 6: State the answer.
\[ \boxed{239} \]