Permutatations vs variations vs combinations
1 when to choose what. Please explain with an example. In a layman words what would be the difference between these. Please explain differences with more examples
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The final lecture of this section, Summary of Combinatorics, summarizes all three concepts:
Additionally, you can refer to the answer in the followig thread:
Layman Explanation: Think of permutations as different "line-ups". Imagine you're lining up for a photo, and every different order of people in that line results in a different photo.
When to use: When the order of items matters.
Example: You're choosing the 1st, 2nd, and 3rd place winners from a race with 10 participants. The person who comes first is distinct from the one who comes second or third.
Using 10 runners: The number of ways to choose 1st, 2nd, and 3rd places is given by permutations of 10 items taken 3 at a time.
Layman Explanation: This is a bit trickier. Variations are like permutations but with a twist. Imagine you have 10 books, but you're only choosing 3 to take on a trip. Each different order you can pack these books is a variation.
When to use: When you're not using all items, but the order of the ones you're using still matters.
Example: Out of 10 books on your shelf, you want to pack 3 in your bag. If the order in which you read them matters (e.g., a three-part series), then the number of ways to pack them in order is given by variations of 10 items taken 3 at a time.
(It's worth noting that, mathematically, permutations and variations overlap when you're choosing a subset of items in order. So, the formulas for them coincide in this scenario.)
Layman Explanation: Combinations are like making "groups" without caring about the order inside the group. Imagine you're making a team; it doesn't matter who joined the team first or last, just who's on the team.
When to use: When you're selecting items, and the order doesn't matter.
Example: Out of 10 friends, you want to invite 3 to watch a movie. It doesn't matter in which order you call them; you just want them there. The number of ways to choose your 3 friends is given by combinations of 10 items taken 3 at a time.
Permutations: Think about the different ways numbers can be arranged in a lottery draw. If you have a lottery where you pick 5 numbers out of 50, and the order matters, that's permutations.
Variations: Imagine picking 3 different toppings for a pizza from a list of 10. The order in which they're added could change the taste slightly (first cheese, then olives, then peppers vs. first olives, then cheese, then peppers).
Combinations: Consider making a fruit salad using 3 fruits out of 7 available. Whether you picked apples first, oranges second, and bananas third, or bananas first, apples second, and oranges third, you'd still have the same three fruits in the salad.
Permutations = specific line-ups or orders.
Variations = specific line-ups from a subset.
Combinations = groups, without caring about the order inside.