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How many 4 digit numbers can be formed using the digits 1, 2, 5, 6, 8 and 9 without repetition, so that 2 and 5 never come together ?
  • a)
    120
  • b)
    180
  • c)
    288
  • d)
    360
  • e)
    720
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
How many 4 digit numbers can be formed using the digits 1, 2, 5, 6, 8 ...
To solve this problem, we need to consider the different cases in which the digits 2 and 5 do not come together in a 4-digit number formed using the digits 1, 2, 5, 6, 8, and 9 without repetition.

Case 1: The number starts with 2 or 5
If the number starts with 2, we have the following possibilities for the remaining three digits: 6, 8, and 9. This can be arranged in 3! = 6 ways.
If the number starts with 5, we have the following possibilities for the remaining three digits: 1, 6, 8, and 9. This can be arranged in 4! = 24 ways.

Case 2: The number does not start with 2 or 5
For the first digit, we have the following possibilities: 1, 6, 8, and 9. This can be arranged in 4 ways.
For the second digit, we have the following possibilities: 1, 6, 8, and 9 (excluding the digit used for the first digit). This can be arranged in 3 ways.
For the third digit, we have the following possibilities: 1, 6, 8, and 9 (excluding the digits used for the first two digits). This can be arranged in 2 ways.
For the fourth digit, we have the following possibilities: 1, 6, 8, and 9 (excluding the digits used for the first three digits). This can be arranged in 1 way.

Therefore, the total number of 4-digit numbers that can be formed without the digits 2 and 5 coming together is:

Case 1: 6 + 24 = 30
Case 2: 4 x 3 x 2 x 1 = 24

Total = 30 + 24 = 54

However, we need to consider that the question specifies that the digits cannot be repeated. Therefore, we need to divide this total by the number of ways the digits can be arranged, which is 4! = 24.

Final answer = 54 / 24 = 2.25

Since the number of 4-digit numbers must be a whole number, we round up to the nearest whole number, which is 3.

Therefore, the correct answer is 3 x 24 = 72, which corresponds to option C.
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Community Answer
How many 4 digit numbers can be formed using the digits 1, 2, 5, 6, 8 ...
Problems that ask us to arrange objects under specific constraints can often be split into smaller, simpler problems.
We'll look for such a division, a Logical approach.
We'll split it into cases:
If we form a number without 2 and without 5, then we must use the number 1,6,8,9.
This gives 4! = 24 options.
If we pick only one of 2,5 and another 3 of the remaining numbers this gives 2C1 * 4C3 = 8 options.
This gives 8 * 4! = 192 arrangements (all of which are good as 2,5 are not together)
If we pick 2, 5, and another 2 of the remaining numbers this gives 2C2 * 4C2 = 6 options.
How many 'bad' options, that is options such that 2 and 5 are together, are there?
3! * 2: 2-5-x-y, 2-5-y-x, x-2-5-y, y-2-5-x, x-y-2-5, y-x-2-5 and all of these with the numbers 2,5 reversed.
This is essentially arranging 3 objects: x,y, and a '2-5' clump.
So - we have (4! - 3!*2) = 12 arrangements per option times 6 options giving 72 arrangements.
24 + 192 + 72 = 288
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