Questionbank: Permutation and Probability

Multiples of 12 have to be multiples of 2 and 6. Can we say that a number that is a multiple of 2 and 6 will be a multiple of 12?

Counting 5 Digit Numbers

    Q.8: From the digits 2, 3, 4, 5, 6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?
    1. 36
    2. 60
    3. 84
    4. 72

 

  • Correct Answer
    Choice B. 60

Detailed Solution

Any multiple of 12 should be a multiple of 4 and 3. First, let us look at the constraint for a number being a multiple of 3. Sum of the digits should be a multiple of 3. Sum of all numbers from 2 to 7 is 27. So, if we have to drop a digit and still retain a multiple of 3, we should drop either 3 or 6.

So, the possible 5 digits are 2, 4, 5, 6, 7 or 2, 3, 4, 5, 7.

When the digits are 2, 4, 5, 6, 7. the last two digits possible for the number to be a multiple of 4 are 24, 64, 52, 72, 56, 76. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 36 different numbers.

When the digits are 2, 3, 4, 5, 7. The last two digits possible for the number to be a multiple of 4 are 32, 52, 72, 24. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 24 different numbers.

Overall, there are 60 different 5-digit numbers possible.

Correct Answer: 60

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