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All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by 5, are arranged in increasing order. Find the 2000th number in this list.
  • a)
    4215673
  • b)
    4316572
  • c)
    4317256
  • d)
    4315672
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, ...
Total number of 7-digit numbers starting with 1 and remaining digits 2, 3, 4, 5, 6, and 7 exactly once is equal to 6! = 720
The number of 7-digits numbers starting with 1 and containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once that are not divisible by 5 = 720 – 120 = 600
Similarly we can find for the 7 digit numbers starting with 2 and 3 containing the given digits exactly once but not divisible by 5 as 600
Therefore, 2000th number should start with the digit 4
The number of 7 digit numbers starting with 41 and 42 that are not divisible by 5 is 96[120-24] each.
Thus 2000th number should start with 43
The next 8 numbers in the list are: 4312567, 4312576, 4312657, 4312756, 4315267, 4315276, 4315627 and 4315672. Thus 2000-th number in the list is 4315672.
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Community Answer
All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, ...
To find the 2000th number in the list of 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once and not divisible by 5, we can follow the steps below:

Step 1: Determine the total number of 7-digit numbers that satisfy the given conditions.
Since there are 7 digits to arrange, the total number of arrangements is 7! = 5040.
However, we need to exclude the numbers that are divisible by 5. Out of the 7 digits, 5 is the only digit that is divisible by 5. So, the total number of numbers divisible by 5 is 6! = 720.
Therefore, the total number of 7-digit numbers not divisible by 5 is 5040 - 720 = 4320.

Step 2: Determine the first digit of the 2000th number.
Since the numbers are arranged in increasing order, we can divide the total number of numbers by the number of digits (7) to get the number of numbers in each digit position.
4320 divided by 7 gives a quotient of 617. The remainder is 1, which means the 2000th number falls in the 1st position of the first digit group.

Step 3: Determine the second digit of the 2000th number.
To determine the second digit, we need to find the position within the 1st digit group. Since there are 6 digits remaining (excluding the first digit), we divide the remaining numbers (4320 - 617 = 3703) by 6 to get the number of numbers in each second digit position.
3703 divided by 6 gives a quotient of 617. The remainder is 1, which means the 2000th number falls in the 1st position of the second digit group.

Step 4: Repeat the process for the remaining digits.
Following the same process, we find that the positions within the remaining digit groups are as follows:
- Third digit: Position 5
- Fourth digit: Position 7
- Fifth digit: Position 6
- Sixth digit: Position 3
- Seventh digit: Position 2

Step 5: Determine the actual digits of the 2000th number.
Since the digits are arranged in increasing order, we can determine the actual digits by arranging the digits 1, 2, 3, 4, 5, 6, 7 in increasing order and selecting the digits at the positions we found in the previous steps.
Arranging the digits in increasing order gives us 1234567.
Selecting the digits at the positions we found gives us:
- First digit: 4
- Second digit: 2
- Third digit: 1
- Fourth digit: 5
- Fifth digit: 6
- Sixth digit: 7
- Seventh digit: 3

Therefore, the 2000th number in the list is 4215673, which corresponds to option A.
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