The number of ways in which we can arrange the digits 1, 2, 3....9 such that the product of five digits at any of the five consecutive

positions is divisible by 7 is

To determine the number of ways to arrange the digits 1 through 9 such that the product of any five consecutive digits is divisible by 7, we first need to identify how many of these digits are divisible by 7. In our case, the only digit from 1 to 9 that meets this criterion is the number 7 itself. This means that for the product of any five consecutive digits to be divisible by 7, at least one of those five digits must be the digit 7.

Understanding the Arrangement

Since we are arranging the digits 1 through 9, we have a total of 9 digits. The key point here is that we need to ensure that the digit 7 appears in every group of five consecutive digits. This requirement will dictate how we can position the digit 7 within the arrangement.

Positioning the Digit 7

Let’s analyze the positions where the digit 7 can be placed. If we denote the positions of the digits as follows:

• Position 1
• Position 2
• Position 3
• Position 4
• Position 5
• Position 6
• Position 7
• Position 8
• Position 9

For the product of any five consecutive digits to be divisible by 7, the digit 7 must occupy one of the first five positions (1 to 5) or the last five positions (5 to 9). This means that the digit 7 can be placed in any of the following positions:

• Position 1
• Position 2
• Position 3
• Position 4
• Position 5
• Position 6
• Position 7
• Position 8
• Position 9

However, if we place the digit 7 in position 5, it will still cover the last five positions (5, 6, 7, 8, 9). Thus, the digit 7 can be placed in any of the 5 positions (1 to 5) to ensure that all groups of five consecutive digits include it.

Calculating the Arrangements

Now that we know where we can place the digit 7, we can calculate the total number of arrangements. Once we place the digit 7 in one of the five valid positions, we have 8 remaining digits (1, 2, 3, 4, 5, 6, 8, 9) to arrange in the remaining 8 positions. The number of ways to arrange these 8 digits is given by 8 factorial (8!).

Final Calculation

The total number of arrangements can be calculated as follows:

• Number of positions for 7: 5
• Arrangements of the remaining 8 digits: 8!

Thus, the total number of arrangements is:

Total arrangements = 5 × 8!

Conclusion

Now, calculating 8! gives us 40320. Therefore, the total number of arrangements is:

Total arrangements = 5 × 40320 = 201600

So, there are 201600 different ways to arrange the digits 1 through 9 such that the product of any five consecutive digits is divisible by 7.