To solve the problem of finding the probability that the product of all digits in a nine-digit number formed using the digits 1, 2, 3, 5, and 7 equals 1920, we can follow these steps:
### Step 1: Understand the Problem
We need to form a nine-digit number using the digits 1, 2, 3, 5, and 7. The product of these digits must equal 1920.
### Step 2: Total Number of Cases
Since we can use each digit multiple times, for each of the 9 positions in the number, we have 5 choices (1, 2, 3, 5, or 7). Therefore, the total number of possible nine-digit combinations is:
\[
\text{Total combinations} = 5^9
\]
### Step 3: Prime Factorization of 1920
Next, we need to find the prime factorization of 1920 to understand which digits can be used to achieve this product.
\[
1920 = 2^7 \times 3^1 \times 5^1
\]
### Step 4: Determine the Digits Needed
From the prime factorization, we see that:
- The digit '2' can contribute to the factor of \(2^7\).
- The digit '3' contributes to \(3^1\).
- The digit '5' contributes to \(5^1\).
### Step 5: Count the Required Digits
To achieve the product of 1920, we need:
- Seven '2's
- One '3'
- One '5'
This gives us a total of 9 digits (7 + 1 + 1 = 9), which fits our requirement for a nine-digit number.
### Step 6: Calculate the Arrangements
The number of ways to arrange these digits (7 '2's, 1 '3', and 1 '5') can be calculated using the formula for permutations of multiset:
\[
\text{Arrangements} = \frac{9!}{7! \times 1! \times 1!}
\]
Calculating this:
\[
9! = 362880 \quad \text{and} \quad 7! = 5040
\]
Thus,
\[
\text{Arrangements} = \frac{362880}{5040 \times 1 \times 1} = \frac{362880}{5040} = 72
\]
### Step 7: Calculate the Probability
Now, we can find the probability \(P\) that the product of the digits equals 1920:
\[
P = \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{72}{5^9}
\]
### Step 8: Final Result
Calculating \(5^9\):
\[
5^9 = 1953125
\]
Thus, the probability is:
\[
P = \frac{72}{1953125}
\]
### Conclusion
The probability that the product of all digits in the nine-digit number equals 1920 is:
\[
\frac{72}{1953125}
\]
