There are 3 oranges, 5 apples and 6 mangoes in a fuit basket (all fruits of same kind are identical). Number of ways in which fruits can be selected from the basket, is

Options

A. 124
B. 125
C. 167
D. 168

Answer

To solve the problem of finding the number of ways to select fruits from a basket containing 3 oranges, 5 apples, and 6 mangoes (where all fruits of the same kind are identical), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of each type of fruit:**
- Oranges: 3
- Apples: 5
- Mangoes: 6

2. **Determine the number of ways to select each type of fruit:**
- For oranges, you can select:
- 0 oranges
- 1 orange
- 2 oranges
- 3 oranges
- This gives us a total of \(3 + 1 = 4\) ways to select oranges.

- For apples, you can select:
- 0 apples
- 1 apple
- 2 apples
- 3 apples
- 4 apples
- 5 apples
- This gives us a total of \(5 + 1 = 6\) ways to select apples.

- For mangoes, you can select:
- 0 mangoes
- 1 mango
- 2 mangoes
- 3 mangoes
- 4 mangoes
- 5 mangoes
- 6 mangoes
- This gives us a total of \(6 + 1 = 7\) ways to select mangoes.

3. **Calculate the total number of selections:**
- The total number of ways to select fruits (including the option of selecting none) is given by multiplying the number of ways for each type:
\[ \text{Total selections} = (4 \text{ ways for oranges}) \times (6 \text{ ways for apples}) \times (7 \text{ ways for mangoes}) = 4 \times 6 \times 7 \]

4. **Perform the multiplication:**
- First, calculate \(4 \times 6 = 24\).
- Then, calculate \(24 \times 7 = 168\).
- So, the total number of selections (including selecting none) is 168.

5. **Subtract the case of selecting no fruits:**
- Since we need to find the number of ways to select at least one fruit, we must subtract the case where no fruits are selected (which is 1 way).
\[ \text{Number of ways to select at least one fruit} = 168 - 1 = 167 \]

6. **Final Answer:**
- Therefore, the number of ways in which fruits can be selected from the basket is **167**.