# Questionbank: Permutation and Probability

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A simple question, which can be solved by having knowledge of both combinatorics and basic principles of inequalities

## Possibler Integer Solutions

Q.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

Choice A. 7

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## Detailed Solution

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.

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