To solve the problem of finding the probability that there are four letters between A and B when the first twelve letters of the alphabet are written down at random, we can follow these steps:
### Step 1: Identify the total number of letters
The first twelve letters of the alphabet are: A, B, C, D, E, F, G, H, I, J, K, L.
### Step 2: Determine the arrangement of A and B
We need to place A and B such that there are exactly four letters between them. This can happen in two ways:
1. A is before B (A _ _ _ _ B)
2. B is before A (B _ _ _ _ A)
### Step 3: Calculate the positions for A and B
If we denote the positions of A and B as follows:
- A occupies position \(i\)
- B occupies position \(i + 5\) (since there are 4 letters in between)
The possible values for \(i\) can be 1, 2, 3, 4, 5, 6, or 7 (since if A is in position 7, B will be in position 12). Thus, there are 7 possible pairs of positions for A and B.
### Step 4: Choose the letters between A and B
After placing A and B, we have to fill the four positions between them with any four letters from the remaining 10 letters (C, D, E, F, G, H, I, J, K, L). The number of ways to choose 4 letters from 10 is given by the combination formula \( \binom{n}{r} \), which is:
\[
\binom{10}{4}
\]
### Step 5: Arrange the letters between A and B
The four letters chosen can be arranged in \(4!\) (factorial of 4) different ways.
### Step 6: Arrange the remaining letters
After placing A, B, and the four letters between them, we have 6 letters left (from the original 12 letters). These 6 letters can be arranged in \(6!\) different ways.
### Step 7: Calculate the total arrangements
The total arrangements where A and B have 4 letters between them can be calculated as:
\[
\text{Total arrangements} = 7 \times \binom{10}{4} \times 4! \times 6!
\]
### Step 8: Calculate the total arrangements of 12 letters
The total arrangements of all 12 letters without any restrictions is:
\[
12!
\]
### Step 9: Calculate the probability
The probability \(P\) that there are exactly four letters between A and B is given by the ratio of the favorable arrangements to the total arrangements:
\[
P = \frac{7 \times \binom{10}{4} \times 4! \times 6!}{12!}
\]
### Step 10: Simplify the expression
Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), we can simplify:
\[
\binom{10}{4} = \frac{10!}{4! \cdot 6!}
\]
Thus, substituting this back into the probability formula, we can simplify further.
### Final Calculation
After performing the calculations, we find:
\[
P = \frac{7 \times \frac{10!}{4! \cdot 6!} \times 4! \times 6!}{12!} = \frac{7 \times 10!}{12!}
\]
This simplifies to:
\[
P = \frac{7}{66}
\]
### Conclusion
The probability that there are four letters between A and B is \( \frac{7}{66} \).
---
