Five digit numbers divisible by 9 are to be formed by using the digits 0, 1, 2, 3, 4, 7, 8 (without repetition). the total number of such numbers that can be formed is

To solve this problem, we need to consider two cases:

1. The first case is when the number starts with 0. Since we can't have a five-digit number starting with 0, we only have 6 digits left to form a four-digit number. The number of ways to arrange these 6 digits is 6P4 = 360. However, not all of these numbers are divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits 1, 2, 3, 4, 7, 8 is 25, which is not divisible by 9. Therefore, in this case, no numbers can be formed.

2. The second case is when the number does not start with 0. We have 6 options for the first digit (1, 2, 3, 4, 7, 8), and then 6 options for the remaining 4 digits (0 and the remaining 5 digits from the first step). The number of ways to arrange these 6 digits is 6P4 = 360. However, we need to subtract the numbers that are not divisible by 9. The sum of the digits 0, 1, 2, 3, 4, 7, 8 is 25, which is not divisible by 9. Therefore, we need to subtract the numbers that have a sum of digits equal to 16 (25 - 9). There are 6 such numbers (7, 8, 1 and any 2 of the remaining 3 digits). The number of ways to arrange these 4 digits is 4P4 = 24. Therefore, in this case, the number of numbers that can be formed is 360 - 24 = 336.

So, the total number of five-digit numbers divisible by 9 that can be formed by using the digits 0, 1, 2, 3, 4, 7, 8 (without repetition) is 0 (from the first case) + 336 (from the second case) = 336.

To solve this problem, we need to consider two cases:

Case 1: The number starts with 0.

In this case, we have 6 remaining digits to choose from for the first position (1, 2, 3, 4, 7, 8). For the remaining 4 positions, we can choose any of the remaining 6 digits. So, the total number of numbers that can be formed in this case is 6 * 5 * 4 * 3 * 2 = 720.

Case 2: The number does not start with 0.

In this case, we have 6 digits to choose from for the first position (1, 2, 3, 4, 7, 8). For the remaining 4 positions, we can choose any of the remaining 6 digits. So, the total number of numbers that can be formed in this case is 6 * 6 * 5 * 4 * 3 = 2160.

However, not all of these numbers are divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits 0, 1, 2, 3, 4, 7, 8 is 25, which is not divisible by 9. Therefore, we need to subtract the numbers that include the digit 7 (since 25 - 7 = 18, which is divisible by 9).

For the numbers that include the digit 7, we have 5 positions to fill. For the first position, we have 5 choices (1, 2, 3, 4, 8). For the remaining 4 positions, we can choose any of the remaining 5 digits. So, the total number of numbers that include the digit 7 is 5 * 5 * 4 * 3 * 2 = 600.

Therefore, the total number of five-digit numbers divisible by 9 that can be formed by using the digits 0, 1, 2, 3, 4, 7, 8 (without repetition) is 2160 + 720 - 600 = 2280.

To solve this problem, we need to understand a few key concepts:

1. A number is divisible by 9 if the sum of its digits is divisible by 9.
2. We are forming a 5-digit number using the digits 0, 1, 2, 3, 4, 7, 8 without repetition.

First, let's add up the digits we have: 0 + 1 + 2 + 3 + 4 + 7 + 8 = 25.

Since 25 is not divisible by 9, we cannot form a 5-digit number that is divisible by 9 using all the digits.

However, we can form a 5-digit number that is divisible by 9 by excluding one of the digits.

We need to find a digit to exclude so that the sum of the remaining digits is divisible by 9.

If we exclude 7, the sum of the remaining digits is 18 (which is divisible by 9): 0 + 1 + 2 + 3 + 4 + 8 = 18.

Now, we need to form a 5-digit number using the digits 0, 1, 2, 3, 4, 8.

The first digit cannot be 0 (since it's a 5-digit number), so we have 5 choices for the first digit.

For the second digit, we have 6 choices (including 0).

For the third, fourth, and fifth digits, we have 5, 4, and 3 choices respectively.

So, the total number of 5-digit numbers divisible by 9 that can be formed is 5 * 6 * 5 * 4 * 3 = 1800.