There are 10 oranges in a basket. Find the no. of ways in which 3 oranges are choosen from the basket?

Options

A. 125
B. 140
C. 110
D. 120

Answer

To find the number of ways to choose 3 oranges from a basket of 10 oranges, we can use the concept of combinations. The formula for combinations is given by:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

where:
- \( n \) is the total number of items (oranges in this case),
- \( r \) is the number of items to choose,
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

### Step-by-Step Solution:

1. **Identify the values of \( n \) and \( r \)**:
- Total oranges \( n = 10 \)
- Oranges to choose \( r = 3 \)

2. **Substitute the values into the combination formula**:
\[ C(10, 3) = \frac{10!}{3!(10-3)!} \]
This simplifies to:
\[ C(10, 3) = \frac{10!}{3! \cdot 7!} \]

3. **Expand the factorials**:
- \( 10! = 10 \times 9 \times 8 \times 7! \)
- Therefore, we can rewrite the equation as:
\[ C(10, 3) = \frac{10 \times 9 \times 8 \times 7!}{3! \cdot 7!} \]

4. **Cancel out \( 7! \)**:
\[ C(10, 3) = \frac{10 \times 9 \times 8}{3!} \]

5. **Calculate \( 3! \)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]

6. **Substitute \( 3! \) back into the equation**:
\[ C(10, 3) = \frac{10 \times 9 \times 8}{6} \]

7. **Perform the multiplication in the numerator**:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]

8. **Divide by \( 6 \)**:
\[ C(10, 3) = \frac{720}{6} = 120 \]

### Final Answer:
The number of ways to choose 3 oranges from a basket of 10 oranges is **120**.

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