How many ways can a subcommittee of three people be selected...

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. This concept is used when we want to form groups (such as a subcommittee) and only care about which items are selected, not the arrangement. The formula for combinations, often written as n choose r, calculates how many groups of r items can be selected from a set of n items without regard to order.

Permutations

Permutations refer to the arrangement of items where the order does matter. This concept applies when selecting individuals for specific positions (such as president, vice-president, and secretary) because each role is distinct. The permutation formula calculates the number of ways to arrange r elements out of a set of n elements, taking into account the order in which they are placed.

Transcript

00:01 We are being asked two different questions on this problem.

00:04 First thing we are asked is how many ways can a subcommittee of three people be selected from a committee of seven people? well, just dealing with that.

00:14 Anytime we're being asked how many ways we can do something, the first thing we want to determine is whether we are dealing with a combination or a permutation.

00:23 Because we know we're going to end up being able to plug stuff into our calculator eventually, using and finding an n value and r value, but those n values and r values are useless to us if we don't know whether we're plugging it in for a combination or permutation.

00:37 The difference between the two, like it says here on the screen, is that in a combination, your order does not matter, whereas in a permutation, your order does matter.

00:49 So if we're thinking about this specific scenario, our first question here, in how many ways can a subcommittee of three people be selected from a committee of seven.

00:57 The question i want to ask herself is, does the order matter? does it matter to those three people, whether they are the first one in the subcommittee or the second one or the third one? in this case, the answer is no.

01:12 We do not care about the order.

01:14 Because if it's a committee and they're just three members, then in theory, they all have an equal amount of input and equal say in the committee.

01:23 Right? it doesn't say that any of them is the head of the committee or anything like that, they're just three people that are trying to work together on whatever this hypothetical issue is.

01:34 So since it doesn't matter to us order -wise, part a, or our first question here, is going to be a combination.

01:42 Now, we can see that there was a total of seven people in the committee, and then we're picking three for this subcommittee, so we had seven people, and we are choosing three of them, so we have our n and r value, but we would be looking to do a combination with those numbers.

02:03 Well, if you do a combination of those numbers in your calculator, and again, remember depending on what calculator you have, you may type in 7 and then your combination with 3.

02:19 Just depends on how your calculator works, but if you do 7, 2, 3 with a combination, you should get 35, meaning there are 35 different ways that we could do that, that we could select three people from a committee of seven.

02:37 Now we compare that, or we should compare that, to what we have for part b...

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