Numbers of ways in which three fruits be selected out of 20 fruits in which 10-mangoes, 5-apples, 2-oranges and rest are different, are

Options

A. 1583
B. 1577
C. 1559
D. None of these

Answer

To solve the problem of selecting 3 fruits from a total of 20 fruits, which includes 10 mangoes, 5 apples, 2 oranges, and the rest being different fruits, we can follow these steps:

### Step 1: Understand the Total Fruits
We have a total of 20 fruits:
- 10 Mangoes
- 5 Apples
- 2 Oranges
- 3 Different Fruits (since 20 - (10 + 5 + 2) = 3)

### Step 2: Identify the Selection Requirement
We need to select 3 fruits from these 20 fruits without any restrictions on the type of fruits selected.

### Step 3: Use the Combination Formula
The number of ways to choose \( r \) objects from \( n \) objects is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
In our case, \( n = 20 \) and \( r = 3 \).

### Step 4: Substitute Values into the Formula
Using the combination formula:
\[ C(20, 3) = \frac{20!}{3!(20 - 3)!} = \frac{20!}{3! \cdot 17!} \]

### Step 5: Simplify the Factorial Expression
We can simplify \( 20! \) as follows:
\[ 20! = 20 \times 19 \times 18 \times 17! \]
Thus, we can cancel \( 17! \) from the numerator and denominator:
\[ C(20, 3) = \frac{20 \times 19 \times 18}{3!} \]

### Step 6: Calculate \( 3! \)
Calculating \( 3! \):
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 7: Substitute Back and Calculate
Now substituting back into the equation:
\[ C(20, 3) = \frac{20 \times 19 \times 18}{6} \]

### Step 8: Perform the Multiplication
Calculating the numerator:
\[ 20 \times 19 = 380 \]
\[ 380 \times 18 = 6840 \]

### Step 9: Divide by 6
Now divide by \( 6 \):
\[ \frac{6840}{6} = 1140 \]

### Final Answer
The number of ways to select 3 fruits from the 20 available fruits is **1140**.

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