how many four digit number are there between 3200 and 7300 in which 6 8 9 together or separately do not appear

And answer is 1077
Please explain in easy way

not too clear on the wording ...

would 45637 be allowed, or do all of the 6 8 9 have to be excluded?

Well, it seems like you're trying to find the number of four-digit numbers between 3200 and 7300 that do not have the digits 6, 8, or 9. To make it easier, let's break it down step by step.

Step 1: Identify the range
We want to find four-digit numbers between 3200 and 7300. So our starting point is 3200, and our ending point is 7300.

Step 2: Define the restrictions
We need to exclude numbers that have the digits 6, 8, or 9, either together or separately. This means any number that contains one or more of these digits can't be included.

Step 3: Count the possibilities
Since our numbers are four-digit numbers, let's start by looking at each place value from left to right: thousands, hundreds, tens, and units.

- Thousands place: Since the given range is between 3200 and 7300, our thousands place options are 3, 4, 5, 6, and 7. But we need to exclude 6, 8, and 9. So our thousands place can only be either 3, 4, 5, or 7. We have 4 options here.

- Hundreds place: For the hundreds place, all digits from 0 to 9 are allowed, except for 6, 8, and 9. So we have 10 - 3 = 7 options here.

- Tens place: The same restrictions apply here, so we also have 7 options for the tens place.

- Units place: Similar to the previous places, we have 7 options for the units place.

Step 4: Multiply the possibilities
To find the total number of four-digit numbers that meet the given conditions, we multiply the number of options for each place value.

4 options × 7 options × 7 options × 7 options = 1072

So, according to my calculations, there are 1072 four-digit numbers between 3200 and 7300 in which the digits 6, 8, and 9 do not appear together or separately.

To find the number of four-digit numbers between 3200 and 7300 in which the digits 6, 8, and 9 do not appear, we can break it down into subproblems.

Step 1: Total number of four-digit numbers between 3200 and 7300:
To find this, we need to subtract the lowest four-digit number (3200) from the largest four-digit number (7300) and add 1 because both numbers are inclusive.
Total number of four-digit numbers = (7300 - 3200) + 1 = 4101.

Step 2: Find the number of four-digit numbers that contain the digit 6, 8, or 9:
Here, we need to consider the three cases separately because the digits 6, 8, and 9 can appear either together or separately in a four-digit number.

Case 1: Numbers that have 6, 8, and 9 together:
There are 10 possible positions for these three digits (e.g., 6_89, _689, 68_9, 689_). For each position, we have 7 choices for the remaining digit (0, 1, 2, 3, 4, 5, or 7). Therefore, the total number of four-digit numbers with 6, 8, and 9 together is 10 * 7 = 70.

Case 2: Numbers that have 6 or 8 alone:
In this case, we have 6 possible positions for the first digit (0 cannot be the first digit of a four-digit number) and 9 choices for each of the remaining three digits (excluding 6, 8, and 9). Therefore, the total number of four-digit numbers with 6 or 8 alone is 6 * 9 * 9 * 9 = 4374.

Case 3: Numbers that have 9 alone:
Similar to the previous case, we have 6 possible positions for the first digit and 9 choices for each of the remaining three digits. Therefore, the total number of four-digit numbers with 9 alone is 6 * 9 * 9 * 9 = 4374.

Step 3: Subtract the numbers from Step 2 from the total number in Step 1:
To get the count of four-digit numbers that do not contain the digits 6, 8, or 9, we subtract the numbers in Step 2 from the total number of four-digit numbers.
Count = Total number - (Numbers with 6, 8, and 9 together + Numbers with 6 or 8 alone + Numbers with 9 alone)
Count = 4101 - (70 + 4374 + 4374) = 1077.

Therefore, there are 1077 four-digit numbers between 3200 and 7300 in which the digits 6, 8, and 9 do not appear.

To find the number of four-digit numbers between 3200 and 7300 in which the digits 6, 8, and 9 do not appear together or separately, we can break down the problem into smaller steps:

Step 1: Determine the total number of four-digit numbers between 3200 and 7300.
To do this, we subtract the smallest number (3200) from the largest number (7300) and then add 1.
7300 - 3200 + 1 = 4101

Step 2: Determine the number of four-digit numbers in which the digits 6, 8, and 9 appear together or separately.
To do this, we need to consider three cases:
Case 1: Digits 6, 8, and 9 appear together.
In this case, we treat 6, 8, and 9 as a single digit and count the possible remaining digits. The remaining digit can be any number from 0 to 5 or from 7 to 9. Therefore, we have 8 choices for the remaining digit. For the thousands place, we have no restriction, so we have 10 choices. So, the total number of four-digit numbers where 6, 8, and 9 appear together is 8 x 10 = 80.

Case 2: Digit 6 appears separately.
In this case, we treat digit 6 as a single digit and count the possible choices for the remaining digits. Each remaining digit can be any number from 0 to 5, 7, or 8. Therefore, we have 8 choices for each of the remaining digits. For the thousands place, we have 9 choices (digits can't be 0). So, the total number of four-digit numbers where 6 appears separately is 8 x 8 x 9 = 576.

Case 3: Digits 8 and 9 appear separately.
In this case, we treat digits 8 and 9 as separate digits and count the possible choices for the remaining digits. Each remaining digit can be any number from 0 to 5 or from 7 to 9. Therefore, we have 8 choices for each of the remaining digits. For the thousands place, we have 9 choices (digits can't be 0). So, the total number of four-digit numbers where 8 and 9 appear separately is 8 x 8 x 9 = 576.

Step 3: Calculate the number of four-digit numbers in which the digits 6, 8, and 9 do not appear together or separately.
To calculate this, we subtract the numbers calculated in Step 2 from the total number of four-digit numbers calculated in Step 1.
4101 - (80 + 576 + 576) = 4101 - 1232 = 2869

Therefore, the number of four-digit numbers between 3200 and 7300 in which the digits 6, 8, and 9 do not appear is 2869.