**What is the difference between ordered and unordered pairs?**

This is a fairly common question. The textbook definition is simple – If the order matters, we are talking about ordered pairs, if order does not matter we are referring to unordered pairs. But this idea probably needs more elaborating than just the definition. Let us do this with an example –

- In how many ways can two natural numbers add up to 15?
- If x and y are natural numbers, how many pairs of values (x, y) that satisfy the equation x + y = 15?

In the first case, we are thinking of combinations 1 + 14, 2 + 13, 3 + 12, 4 + 11, 5 + 10, 6 + 9, 7 + 8 – there are seven different ways.

In the second case, we are thinking of x = 1, y = 14, x = 2, y = 13 and so on. x = 14 and y = 1 will also be counted, so will x = 13, y = 2. Some questions will explicitly ask for number of ordered pairs (x, y) that satisfy the equation x + y = 15, but even as a general rule if we are asked for pairs of values x, y then we should count x = a, y = b and x =b, y = a if both satisfy the given conditions.

**Let us have another example **

- Sum of three natural numbers is 5. In how many ways can this be done?
- Or, a + b + c = 5 – if a, b, c are natural numbers, how many ordered triplets exist?

Sum of three natural numbers is 5. This could be 2 + 2 + 1 or 3 + 1 + 1. Two ways. This is the answer for the first question.

For the second one, there are six possibilities. For 2, 2, 1, we could have a = 2, b = 2, c = 1, a = 2, b = 1, c = 2 or a = 1,b =2 , c =2. Similarly there are three variants for 3, 1,1 – {(3,1,1) , (1, 3, 1), (1, 1, 3)}

**Don’t talk about ordered and unordered when the question has nothing to do about it **

Question: How many solutions are there for the equation 2x + 5y = 103 is x and y are both natural numbers?

In this question, *please* do not ask whether we are talking about ordered or unordered pairs. This question has nothing to do with ordered or unordered pairs.

x = 4, y = 95 works. x = 95 and y = 4 does not work. In this question **x and y are not interchangeable**. So, there is no question of ordered/unordered pairs here.

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