A small pack of cards consists of 5 green cards 4 blue cards and 3 black cards. The pack is shuffled through and first three cards are turned face up. The probability that there is exactly one card of each colour is

Options

A. `(9)/(55)`
B. `(4)/(11)`
C. `(3)/(11)`
D. `(8)/(55)`

Answer

To solve the problem of finding the probability that there is exactly one card of each color when three cards are drawn from a pack containing 5 green cards, 4 blue cards, and 3 black cards, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Number of Cards**:
- We have 5 green cards, 4 blue cards, and 3 black cards.
- Total number of cards = 5 + 4 + 3 = 12.

2. **Calculate the Total Ways to Choose 3 Cards from 12**:
- The total number of ways to choose 3 cards from 12 is given by the combination formula \( \binom{n}{r} \), which is \( \frac{n!}{r!(n-r)!} \).
- Here, \( n = 12 \) and \( r = 3 \):
\[ \text{Total ways} = \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220. \]

3. **Calculate the Ways to Choose One Card of Each Color**:
- We need to choose 1 card from each color: 1 green, 1 blue, and 1 black.
- The number of ways to choose 1 green card from 5 is \( \binom{5}{1} = 5 \).
- The number of ways to choose 1 blue card from 4 is \( \binom{4}{1} = 4 \).
- The number of ways to choose 1 black card from 3 is \( \binom{3}{1} = 3 \).
- Therefore, the total ways to choose one card of each color is:
\[ \text{Ways to choose} = 5 \times 4 \times 3 = 60. \]

4. **Calculate the Probability**:
- The probability of drawing exactly one card of each color is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{60}{220}. \]
- Simplifying this fraction:
\[ \frac{60}{220} = \frac{3}{11}. \]

### Final Answer:
The probability that there is exactly one card of each color is \( \frac{3}{11} \).