Combinatorial principles
Okay, let's imagine you have a bowl of strawberries and a bowl of blueberries. You're going to make a dessert, and you want to use a combination of both fruits. Combinatorial principles are like the rules you can use to figure out how many different desserts you can make.
One thing to consider is how many options you have for each ingredient. You have 10 strawberries and 8 blueberries, so you have 10 choices for the first fruit and 8 choices for the second.
Another thing to consider is how many ingredients you want to use. For example, you might want to use 1 strawberry and 1 blueberry, or you might want to use 2 strawberries and 3 blueberries. The rules for figuring out these combinations are called permutations and combinations.
Permutations are like the order of the ingredients matters. So if you make a dessert with 1 strawberry and 1 blueberry, you can make a strawberry-blueberry dessert or a blueberry-strawberry dessert. Those are two different possibilities. The formula for permutations is:
nPr = n! / (n-r)!
where n is the number of options you have (10 strawberries + 8 blueberries = 18), and r is the number of ingredients you're using (2 in this case).
So if you want to know how many different 2-ingredient desserts you can make using strawberries and blueberries, you would use the permutation formula like this:
nPr = 18! / (18-2)! = 18! / 16! = 18 x 17 = 306
So you have 306 different options for your dessert!
Combinations are like the order of the ingredients doesn't matter. So if you make a dessert with 1 strawberry and 1 blueberry, it doesn't matter what order you put them in. The formula for combinations is:
nCr = n! / (r! (n-r)!)
So if you want to know how many different 2-ingredient desserts you can make using strawberries and blueberries, but you don't care about the order, you would use the combination formula like this:
nCr = 18! / (2! (18-2)!) = 18! / (2! 16!) = (18 x 17) / 2 = 153
So you have 153 different options for your dessert if you don't care about the order.
So combinatorial principles just help you figure out how many different combinations of things you can make when you have different options to choose from. It's like a big math game where you get to mix and match things however you want!
One thing to consider is how many options you have for each ingredient. You have 10 strawberries and 8 blueberries, so you have 10 choices for the first fruit and 8 choices for the second.
Another thing to consider is how many ingredients you want to use. For example, you might want to use 1 strawberry and 1 blueberry, or you might want to use 2 strawberries and 3 blueberries. The rules for figuring out these combinations are called permutations and combinations.
Permutations are like the order of the ingredients matters. So if you make a dessert with 1 strawberry and 1 blueberry, you can make a strawberry-blueberry dessert or a blueberry-strawberry dessert. Those are two different possibilities. The formula for permutations is:
nPr = n! / (n-r)!
where n is the number of options you have (10 strawberries + 8 blueberries = 18), and r is the number of ingredients you're using (2 in this case).
So if you want to know how many different 2-ingredient desserts you can make using strawberries and blueberries, you would use the permutation formula like this:
nPr = 18! / (18-2)! = 18! / 16! = 18 x 17 = 306
So you have 306 different options for your dessert!
Combinations are like the order of the ingredients doesn't matter. So if you make a dessert with 1 strawberry and 1 blueberry, it doesn't matter what order you put them in. The formula for combinations is:
nCr = n! / (r! (n-r)!)
So if you want to know how many different 2-ingredient desserts you can make using strawberries and blueberries, but you don't care about the order, you would use the combination formula like this:
nCr = 18! / (2! (18-2)!) = 18! / (2! 16!) = (18 x 17) / 2 = 153
So you have 153 different options for your dessert if you don't care about the order.
So combinatorial principles just help you figure out how many different combinations of things you can make when you have different options to choose from. It's like a big math game where you get to mix and match things however you want!