A box contains 6 distinct dolls. From this box, three dolls are randomly selected one by one with replacement. What is the probability of selecting 3 distinct dolls?

Options

A. `5//54`
B. `12//25`
C. `1//20`
D. `5//9`

Answer

To solve the problem of finding the probability of selecting 3 distinct dolls from a box containing 6 distinct dolls, where the selection is done one by one with replacement, we can follow these steps:

### Step 1: Understand the Selection Process
Since we are selecting dolls with replacement, the total number of dolls remains the same for each selection. There are 6 distinct dolls available.

### Step 2: Calculate the Total Outcomes
For each selection, there are 6 possible outcomes (since there are 6 distinct dolls). Therefore, when selecting 3 dolls, the total number of possible outcomes is:
\[ \text{Total outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 3: Calculate the Favorable Outcomes for Distinct Dolls
To find the number of favorable outcomes where all 3 selected dolls are distinct, we can consider the following:
- For the first doll, we have 6 choices.
- For the second doll, we must choose a different doll, so we have 5 choices.
- For the third doll, we must choose a different doll from the first two, leaving us with 4 choices.

Thus, the number of favorable outcomes is:
\[ \text{Favorable outcomes} = 6 \times 5 \times 4 = 120 \]

### Step 4: Calculate the Probability
The probability of selecting 3 distinct dolls is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{120}{216} \]

### Step 5: Simplify the Probability
To simplify \(\frac{120}{216}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 24:
\[ \frac{120 \div 24}{216 \div 24} = \frac{5}{9} \]

### Final Answer
Thus, the probability of selecting 3 distinct dolls is:
\[ \frac{5}{9} \]

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