A bag has four pair of balls of four distinct colours. If four balls are picked at random (without replacement), the probability that there is atleast one pair among them have the same colour is

Options

A. `(1)/(7!)`
B. `(8)/(35)`
C. `(19)/(35)`
D. `(27)/(35)`

Explanation

To solve the problem, we need to find the probability that when picking 4 balls from a bag containing 4 pairs of balls of 4 distinct colors, there is at least one pair among them that has the same color.

### Step-by-Step Solution:

1. **Identify the Total Number of Balls:**
- There are 4 pairs of balls, which means there are \( 4 \times 2 = 8 \) balls in total.

2. **Calculate the Total Ways to Choose 4 Balls:**
- The total number of ways to choose 4 balls from 8 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total ways} = \binom{8}{4} \]

3. **Calculate the Number of Ways to Choose 4 Balls with No Same Color:**
- To ensure that no two balls have the same color, we can only select one ball from each color. Since there are 4 colors, we can choose 1 ball from each of the 4 colors.
- The number of ways to choose 1 ball from each of the 4 pairs is:
\[ \text{Ways with no same color} = 2^4 = 16 \]
(since for each color, we have 2 choices).

4. **Calculate the Probability of Choosing Balls with No Same Color:**
- The probability of choosing 4 balls such that none of them have the same color is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{no same color}) = \frac{\text{Ways with no same color}}{\text{Total ways}} = \frac{16}{\binom{8}{4}} \]

5. **Calculate \( \binom{8}{4} \):**
- We compute \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

6. **Substitute Back to Find the Probability:**
- Now substituting back into the probability formula:
\[ P(\text{no same color}) = \frac{16}{70} = \frac{8}{35} \]

7. **Find the Probability of Having At Least One Pair:**
- The probability of having at least one pair is the complement of the probability of having no pairs:
\[ P(\text{at least one pair}) = 1 - P(\text{no same color}) = 1 - \frac{8}{35} = \frac{35 - 8}{35} = \frac{27}{35} \]

### Final Answer:
The probability that there is at least one pair among the 4 balls picked is \( \frac{27}{35} \).