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 number of choices for each type of fruit.
- For apples: Since there are 6 identical apples, we can choose from 0 to 6 apples. This gives us 7 options (0, 1, 2, 3, 4, 5, 6).
- For mangoes: There are 3 identical mangoes, so we can choose from 0 to 3 mangoes. This gives us 4 options (0, 1, 2, 3).
- For bananas: There are 2 identical bananas, so we can choose from 0 to 2 bananas. This gives us 3 options (0, 1, 2).

### Step 2: Calculate the total number of combinations.
The total number of ways to choose fruits (including the option of choosing none) can be calculated by multiplying the number of choices for each type of fruit:
\[ \text{Total combinations} = (\text{Choices for apples}) \times (\text{Choices for mangoes}) \times (\text{Choices for bananas}) = 7 \times 4 \times 3 \]
Calculating this gives:
\[ 7 \times 4 = 28 \]
\[ 28 \times 3 = 84 \]

### Step 3: Exclude the case where no fruit is selected.
Since we need to select at least one fruit, we must exclude the case where we choose 0 apples, 0 mangoes, and 0 bananas. This is just one case:
\[ \text{Case of no fruit} = 1 \]

### Step 4: Calculate the final answer.
To find the number of ways to select at least one fruit, we subtract the case of selecting no fruit from the total combinations:
\[ \text{Ways to select at least one fruit} = \text{Total combinations} - \text{Case of no fruit} = 84 - 1 = 83 \]

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

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