Double counting (proof technique)

Double counting is a way of proving things by counting them twice in different ways. Imagine you have some toys in a box. You can count how many toys are in the box by looking at them one by one. This is the first way to count.

Now, imagine instead of counting the toys one by one, you count how many toys are in the box by counting how many of each color there are. For example, you might say there are 3 red toys and 4 blue toys. This is the second way to count.

Double counting means you do both of these ways of counting and use the results to prove something. For example, you might want to prove that there are 7 toys in the box. You can count them one by one and get 7. You can also count the number of each color and add them together to get 7. Since both ways give you the same answer, you know that there are 7 toys in the box.

Double counting is a useful proof technique because it allows you to check your answer in more than one way. This can help you catch mistakes and make sure you are right.