# CAT Quantitative Aptitude Questions | CAT Permutation and Combination Questions

###### CAT Questions | Permutation Probability | Possible Integer Solutions CAT Questions

The question is from Permutation and Combination. A question on possible values. We need to find the number of possible positive integers that satisfies the equation. 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 17: 2a + 5b = 103. How many pairs of positive integer values can a, b take such that a > b?

1. 7
2. 9
3. 14
4. 15

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##### Method of solving this CAT Question from Permutation and Combination: A simple question, which can be solved by having knowledge of both combinatorics and basic principles of inequalities

Let us find the one pair of values for a, b.
a = 4, b = 19 satisfies this equation.
2*4 + 5*19 = 103.

Now, if we increase ‘a’ by 5 and decrease ‘b’ by 2 we should get the next set of numbers. We can keep repeating this to get all values.
Let us think about why we increase ‘a’ by 5 and decrease b by 2.
a = 4, b = 19 works.

Let us say, we increase ‘a’ by n, then the increase would be 2n.
This has to be offset by a corresponding decrease in b.
Let us say we decrease b by ‘m’.
This would result in a net drop of 5m.
In order for the total to be same, 2n should be equal to 5m.
The smallest value of m, n for this to work would be 2, 5.

a = 4, b = 19
a = 9, b = 17
a = 14, b = 15
..
And so on till
a = 49, b = 1
We are also told that ‘a’ should be greater than ‘b’, then we have all combinations from (19, 13) … (49, 1).
7 pairs totally.

The question is "2a + 5b = 103. How many pairs of positive integer values can a, b take such that a > b?"

##### Hence the answer is "7"

Choice A is the correct answer.

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