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

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.

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.

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 Okay, so we're asked, um, how many ways can a photographer and a wedding arrange six people in a row from a group of 10 people? so we have a group of 10 people.

00:14 Wonder how many ways can six of those people be arranged in a room? um, if, well, um right.

00:26 Another bride and the groom are among these 10 people.

00:29 And for a were asked, well, if the bride must be in the picture.

00:37 So, um, since the bride must be right in our chosen group, we consider that as our first person, that must be one of these six people.

00:50 Okay, so the second person.

00:53 Well, then there are nine ways to choose because while there's 10 people so excluding the pride for the second person, there are nine ways to choose.

01:04 And then likewise for the third person or there.

01:06 Now, there are eight ways to choose.

01:09 So continuing in the same way to choose to five persons for the group.

01:16 Right, because one of the people have to be the bride.

01:18 So the choose the other, um, five rights is the bride is already included.

01:24 Well, there exists one times now, you know, times eight times.

01:31 Seven times, six times five, um, waste, which is equal to 15,000 120.

01:44 Okay, now, since the bride can be placed in any of the six positions along the row with the 15,120 um, possible number of groups for the six people is just sex times 15,120 which is equal to and 90,000.

02:08 Um, 720.

02:13 So, if party, right.

02:14 If the brian must be in the picture, how many ways are there toward range? six people in a rope from a tent.

02:22 There are 90,000, 720 ways.

02:28 No, for part b.

02:31 Well, part b were asked, um, if both the bride and groom must be in the pictures be we want what if both, um, if both the bridegroom so the b and g, the bride and groom must be in the picture.

02:55 Must be in the pick cure.

02:59 Okay.

03:00 Um so? well, since the bride and groom must be in the children group, we consider that for choosing the first and second person, right, because they must be in the picture.

03:14 So, um, then really going? consider what? for the third person.

03:20 Then there are eight ways to choose, right? because i began this out of a group of 10.

03:25 So for the third person, so four...

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