From 5 apples, 4 mangoes & 3 bananas, in how many ways we can select atleast two fruits of each variety if fruits of same species are identical?

Answer

To solve the problem of selecting at least two fruits of each variety from 5 apples, 4 mangoes, and 3 bananas, where fruits of the same species are identical, we can follow these steps:

### Step 1: Determine the number of ways to select apples
We need to select at least 2 apples from 5 apples. The possible selections are:
- 2 apples
- 3 apples
- 4 apples
- 5 apples

The number of ways to select apples can be calculated as follows:
- Selecting 2 apples: 1 way (since all apples are identical)
- Selecting 3 apples: 1 way
- Selecting 4 apples: 1 way
- Selecting 5 apples: 1 way

Thus, the total number of ways to select apples is:
\[ 1 + 1 + 1 + 1 = 4 \text{ ways} \]

### Step 2: Determine the number of ways to select mangoes
Next, we need to select at least 2 mangoes from 4 mangoes. The possible selections are:
- 2 mangoes
- 3 mangoes
- 4 mangoes

The number of ways to select mangoes can be calculated as follows:
- Selecting 2 mangoes: 1 way
- Selecting 3 mangoes: 1 way
- Selecting 4 mangoes: 1 way

Thus, the total number of ways to select mangoes is:
\[ 1 + 1 + 1 = 3 \text{ ways} \]

### Step 3: Determine the number of ways to select bananas
Finally, we need to select at least 2 bananas from 3 bananas. The possible selections are:
- 2 bananas
- 3 bananas

The number of ways to select bananas can be calculated as follows:
- Selecting 2 bananas: 1 way
- Selecting 3 bananas: 1 way

Thus, the total number of ways to select bananas is:
\[ 1 + 1 = 2 \text{ ways} \]

### Step 4: Calculate the total number of ways to select the fruits
Now, we can find the total number of ways to select at least 2 fruits of each variety by multiplying the number of ways to select each type of fruit:
\[ \text{Total ways} = (\text{Ways to select apples}) \times (\text{Ways to select mangoes}) \times (\text{Ways to select bananas}) \]
\[ \text{Total ways} = 4 \times 3 \times 2 = 24 \text{ ways} \]

### Final Answer
Thus, the total number of ways to select at least two fruits of each variety is **24 ways**.

---