Seven digit numbers are formed using digits 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition. The probability of selecting a number such that product of any 5 consecutive digits is divisible by either 5 or 7 is P. Then 12P is equal to

Answer

To solve the problem, we need to find the probability \( P \) that a randomly selected seven-digit number formed using the digits 1 to 9 (without repetition) has the property that the product of any five consecutive digits is divisible by either 5 or 7. Then, we will compute \( 12P \).

### Step-by-Step Solution:

1. **Total Seven-Digit Numbers**:
The total number of seven-digit numbers that can be formed using the digits 1 to 9 without repetition is given by:
\[ \text{Total cases} = \binom{9}{7} \times 7! \]
Here, \( \binom{9}{7} \) represents the number of ways to choose 7 digits from 9, and \( 7! \) represents the number of ways to arrange these 7 digits.

2. **Calculating Total Cases**:
\[ \binom{9}{7} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
Therefore, the total number of seven-digit numbers is:
\[ \text{Total cases} = 36 \times 7! = 36 \times 5040 = 181440 \]

3. **Favorable Cases**:
We need to find the number of favorable cases where the product of any five consecutive digits is divisible by either 5 or 7.

- **Divisibility by 5**: The digit 5 must be included in the selection of 7 digits.
- **Divisibility by 7**: The digit 7 must also be included in the selection of 7 digits.

If we exclude both 5 and 7, we have the digits {1, 2, 3, 4, 6, 8, 9} left, which gives us 7 digits.

4. **Counting Cases Excluding 5 and 7**:
The number of ways to select 7 digits excluding 5 and 7 is:
\[ \text{Ways to select 7 digits without 5 and 7} = \binom{7}{7} \times 7! = 1 \times 5040 = 5040 \]

5. **Probability Calculation**:
The probability \( P \) that the product of any five consecutive digits is divisible by either 5 or 7 is given by:
\[ P = 1 - \frac{\text{Number of cases not divisible by 5 or 7}}{\text{Total cases}} = 1 - \frac{5040}{181440} \]
Simplifying this:
\[ P = 1 - \frac{1}{36} = \frac{35}{36} \]

6. **Calculating \( 12P \)**:
Now, we calculate \( 12P \):
\[ 12P = 12 \times \frac{35}{36} = \frac{420}{36} = \frac{35}{3} \]

### Final Answer:
Thus, the value of \( 12P \) is:
\[ \boxed{35} \]