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 how many ways fruits can be selected 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, since there are 3 identical oranges, the number of ways to select oranges is \(3 + 1 = 4\). (We can select 0, 1, 2, or 3 oranges)
- For apples, since there are 5 identical apples, the number of ways to select apples is \(5 + 1 = 6\). (We can select 0, 1, 2, 3, 4, or 5 apples)
- For mangoes, since there are 6 identical mangoes, the number of ways to select mangoes is \(6 + 1 = 7\). (We can select 0, 1, 2, 3, 4, 5, or 6 mangoes)

3. **Calculate the total number of ways to select fruits:**
- The total number of ways to select fruits from the basket is the product of the number of ways to select each type of fruit:
\[ \text{Total ways} = (3 + 1) \times (5 + 1) \times (6 + 1) = 4 \times 6 \times 7 \]

4. **Compute the product:**
- Calculate \(4 \times 6 = 24\)
- Then calculate \(24 \times 7 = 168\)

5. **Adjust for the case where no fruits are selected:**
- Since we want the number of ways to select fruits (not including the case where no fruits are selected), we subtract 1 from the total:
\[ \text{Required number of ways} = 168 - 1 = 167 \]

### Final Answer:
The number of ways in which fruits can be selected from the basket is **167**.

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