In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 pe

In how many ways can a photographer at a wedding arrange 6...

Key Concepts

Counting with Restrictions

This concept pertains to modifying counting techniques to account for specific conditions that must be met, such as including or excluding certain individuals. Restrictions may require first fixing the required elements, then applying the standard counting formulas to the remaining selections, ensuring that the criteria are strictly followed.

Inclusion-Exclusion Principle

This is a combinatorial method used to accurately count the number of elements that satisfy multiple conditions, particularly when there is overlap between conditions. It helps avoid overcounting by adding counts of individual cases and then subtracting the counts where conditions intersect, a useful approach when handling cases like choosing exactly one from two specific individuals.

Permutations

This concept involves arranging a set of objects in a specific order, where the sequence matters. In problems like the one described, once a group of people is selected, the goal is to determine how many ordered arrangements (or sequences) can be created from that group, often calculated using the permutation formula nPk = n!/(n-k)!.

Combinations

This concept involves selecting a subset of objects from a larger set where the order of selection does not matter. In these types of problems, combinations are used to choose the individuals for inclusion before arranging them, typically employing the formula nCk = n!/(k!(n-k)!).

Transcript

00:01 For this problem, we are asked, in how many ways can a photographer at a wedding arrange six people in a row from a group of ten people, where the bride and groom are among these ten people? if a, the bride must be in the picture, b, both the bride and groom must be in the picture, or c, exactly one of the bride and groom is in the picture.

00:19 So, for part a, i'd say that we want to think about this essentially as two tasks.

00:26 Task one is picking who's in the picture, and task two is arranging.

00:45 Now, since we have task one and task two, ultimately we can apply the multiplication rule, the product rule, and say that the total number of ways will be the number of ways for task one, multiplied by the number of ways for task two, but we don't know what those will be just yet.

01:04 So for task one.

01:09 We would initially think, okay, we have ten people, and we need to choose six.

01:14 But we were told that the bride and groom are counted amongst those ten.

01:20 So, in part a, we know that we are including the bride.

01:26 So we don't, we're not choosing five, or excuse me, we're not choosing six people.

01:32 One of them is already chosen for us.

01:35 We're only choosing five people.

01:37 So, we can think about it as we have one slot already dedicated to the bride, and then we have five remaining slots we need to fill.

01:52 And we would have nine ways to fill the first slot, eight to fill the second, seven to fill the third, six to fill the fifth, or sorry, fourth, and five to fill the fifth.

02:09 So we'll have nine times eight times seven times six times.

02:13 Five different ways of choosing who out of the original group of 10 will be seating.

02:23 Thing is that in this step, at least, the arrangement doesn't matter.

02:32 Doesn't matter which order we're choosing people.

02:35 Currently, we're just saying, okay, everybody, you, you, you, you're all coming over here for the picture.

02:41 It doesn't matter what order we say that in.

02:45 The number of actual ways here is going to be nine.

02:53 So actually, i'll say ahead of time.

02:56 What we know then is that we're going to have multiple different ways of arriving at the same group for the picture.

03:03 So we know that we'd have nine times eight times seven times six times five different ways of choosing who will be.

03:16 Included and then we need to divide out the number of different ways that this group, this sort of pre -group, we could call it, we need to divide out the number of ways that this pre -group could be arranged.

03:30 So we would have five ways of choosing the first person, and four ways of choosing the second person, and three ways of choosing the third person, two ways of choosing the fourth person, and one way of choosing the fifth person.

03:44 So, the number for our first task is going to be 126.

03:52 For the second task, we are arranging our group of six.

04:02 So, for this, we have, you know, we're arranging them in a row, and we're not told at all about who needs to sit where.

04:15 So we'd have six ways of choosing the first person, five ways of choosing the fourth person.

04:21 Sorry, second person.

04:23 Four ways of choosing the third person.

04:25 Three ways of choosing the fourth person...

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