The probability that the birth days of six different persons will fall in exactly two calendar months is

Options

A. `1/6`
B. `""^(12)C_(2)xx2^(6)/12^(6)`
C. `""^(12)C_(2)xx(2^(6)-1)/12^(6)`
D. `341/12^(5)`

Answer

To solve the problem of finding the probability that the birthdays of six different persons will fall in exactly two calendar months, we can follow these steps:

### Step 1: Choose 2 Months from 12
We need to select 2 months out of the 12 available months in a year. The number of ways to choose 2 months from 12 is given by the combination formula:

\[ \text{Number of ways to choose 2 months} = \binom{12}{2} \]

### Step 2: Assign Birthdays to the Chosen Months
Once we have chosen the 2 months, we need to assign the birthdays of the 6 persons to these months. Each person has 2 choices (either of the two months). Therefore, the total number of ways to assign the birthdays is:

\[ \text{Total ways to assign birthdays} = 2^6 \]

### Step 3: Exclude the Cases Where All Birthdays Fall in One Month
However, we need to exclude the cases where all birthdays fall in only one of the two chosen months. There are 2 such cases (all in the first month or all in the second month). So, the adjusted total ways to assign the birthdays is:

\[ \text{Adjusted total ways} = 2^6 - 2 \]

### Step 4: Calculate the Total Favorable Outcomes
Now, the total number of favorable outcomes (where the birthdays fall in exactly 2 months) is:

\[ \text{Total favorable outcomes} = \binom{12}{2} \times (2^6 - 2) \]

### Step 5: Calculate the Total Possible Outcomes
The total possible outcomes for the birthdays of 6 persons, where each can fall in any of the 12 months, is:

\[ \text{Total possible outcomes} = 12^6 \]

### Step 6: Calculate the Probability
Finally, the probability that the birthdays of the six different persons will fall in exactly two calendar months is given by the ratio of favorable outcomes to total outcomes:

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}} = \frac{\binom{12}{2} \times (2^6 - 2)}{12^6} \]

### Step 7: Simplify the Expression
Now we can simplify the expression:

1. Calculate \(\binom{12}{2} = \frac{12 \times 11}{2} = 66\).
2. Calculate \(2^6 - 2 = 64 - 2 = 62\).
3. Calculate \(12^6 = 2985984\).

Thus, the probability becomes:

\[ \text{Probability} = \frac{66 \times 62}{2985984} \]

### Final Calculation
Calculating the numerator:

\[ 66 \times 62 = 4092 \]

So the probability is:

\[ \text{Probability} = \frac{4092}{2985984} \]

This can be further simplified if needed.

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