There are 11 fruits in a basket of which 6 are apples, 3 mangoes and 2 bananas (fruits of same species are identical). How many ways are there to select atleast one fruit?

Answer

To solve the problem of selecting at least one fruit from a basket containing 6 apples, 3 mangoes, and 2 bananas, we can follow these steps:

### Step 1: Determine the choices for each type of fruit
- For apples (denoted as α), since there are 6 apples, the possible choices for selecting apples are:
- 0, 1, 2, 3, 4, 5, or 6 apples.
- This gives us a total of **7 choices** (including the choice of selecting 0 apples).

- For mangoes (denoted as β), since there are 3 mangoes, the possible choices are:
- 0, 1, 2, or 3 mangoes.
- This gives us a total of **4 choices**.

- For bananas (denoted as γ), since there are 2 bananas, the possible choices are:
- 0, 1, or 2 bananas.
- This gives us a total of **3 choices**.

### Step 2: Calculate the total combinations including the case of selecting no fruits
Now, we can calculate the total number of ways to select fruits (including the case where no fruits are selected):
- Total combinations = (Choices for apples) × (Choices for mangoes) × (Choices for bananas)
- Total combinations = 7 × 4 × 3

Calculating this:
- Total combinations = 7 × 4 = 28
- Total combinations = 28 × 3 = 84

### Step 3: Exclude the case where no fruits are selected
Since the question specifies that we need to select at least one fruit, we need to subtract the case where no fruits are selected (which is 1 case):
- Total ways to select at least one fruit = Total combinations - 1
- Total ways to select at least one fruit = 84 - 1 = 83

### Final Answer
Thus, the total number of ways to select at least one fruit from the basket is **83**.

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