To find the number of pairs of positive integer values of two variables, a and b such that one is greater than the other, under the given constraints/equation.

1. Combinatorics Q9

2. Qn: Combinatorics
2a + 5b = 103. How many pairs of positive integer values can a, b take
such that a > b?
(a) 7 (b) 9
(c) 14 (d) 15

3. Soln: Combinatorics
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.
2a + 5b = 103. How many pairs of positive integer values can a, b take
such that a > b?

4. Soln: Combinatorics
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
2a + 5b = 103. How many pairs of positive integer values can a, b take
such that a > b?

5. Soln: Combinatorics
..
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.
Answer choice (a)
2a + 5b = 103. How many pairs of positive integer values can a, b take
such that a > b?