CAT Quantitative Aptitude Questions | CAT Permutation and Combination Questions

CAT Questions | Permutation Probability | Digit Numbers

The question is from Permutation and Combination. Another question which combines number theory with combinatorics. We need to find out the number of 5 digits which are multiple of 15 and satisfies the given conditions. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 11: How many numbers of up to 5 digits can be created using the digits 1, 2, 3 and 5 each at least once such that they are a multiple of 15?

  1. 24
  2. 18
  3. 15
  4. 12


Video Explanation


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Explanatory Answer

Method of solving this CAT Question from Permutation and Combination: The best combinatorics questions are the ones that involve a bit of Number Systems!

For a number to be a multiple of 15, it has to be a multiple of 3 and of 5. So, the last digit has to be 5 and the sum of digits should be a multiple of 3.
We can have either 4โ€“digit or 5โ€“digit numbers. If we have a 4โ€“digit number, sum of the digits will be 1 + 2 + 3 + 5 = 11.
No 4โ€“digit number formed with digits 1, 2, 3, 5 exactly once can be a multiple of 3. So, there is no possible 4โ€“digit number.

Now, in any 5 digit number, we will have 1, 2, 3, 5 once and one of these 4 digits repeating once. 1 + 2 + 3 + 5 = 11. So, the digit that repeats in order for the number to be a multiple of 3 has to be 1. In this instance, sum of the digits will be 12 and this is the only possibility.
So, any 5โ€“digit number has to have the digits 1, 1, 2, 3, 5. For the number to be a multiple of 5, it has to end in 5.

So, number should be of the form __ __ __ __ 5, with the first 4 slots taken up by 1, 1, 2, 3. These can be rearranged in \\frac{4!}{2!}\\) = 12 ways.
There are 12 possibilities overall.

The question is "How many numbers of up to 5 digits can be created using the digits 1, 2, 3 and 5 each at least once such that they are a multiple of 15?"

Hence the answer is "12"

Choice D is the correct answer.

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