Two camp counselors take 5 kids to the movies and sit in a row of 7 seats. if the counselors must sit in consecutive seats (in either order), how many seating arrangements are possible?

Since the counselors must sit together in any order, first treat them as a single option. This gives 6! possible arrangements for all the participants.

Also, If they can sit in any order, then the total possible arrangements = 2(6!)

= 2( 6 x 5 x 4 x 3 x 2 x 1)

= 1,440 ways

Therefore, the total number of arrangements possible is 1,440 ways

astha8579 astha8579 · Answer 2 · 2022-02-22T13:35:28+01:00

Seating arrangement is unique way in which people can sit. The number of seating arrangements possible in this case is 2520

What is the rule of product in combinatorics?

If a work A can be done in p ways, and another work B can be done in q ways, then both A and B can be done in [tex]p \times q[/tex] ways.

Remember that this count doesn't differentiate between order of doing A first or B first then doing other work after the first work.

Thus, doing A then B is considered same as doing B then A

How to find the number of seating arrangements?

In such situations, we need to model the situation with the view point which can be evaluated mathematically.

For give case, we can see that there are in total 7 seats. And 5 kids are to sit on them, with 2 camp counselors.

So 7 people have to sit on 7 seats.

But it is given that two counselors must sit together.

Now firstly, two counselors can choose 2 seats out of 7 seats in [tex]^7C_2 = \dfrac{7 \times 6}{2 \times 1} = 21[/tex] ways.

Then , in the rest of the 5 seats, 5 kids can arrange themselves in 5! ways(using permutations).

We have:

[tex]n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1\\\\5! =5\times 4\times 3\times 2\times 1 = 120[/tex]

Since each of this 120 arrangement is for each of 21 ways of counselors sitting, thus, there are 120 times 21 ways of those 7 people to sit (using rule of product), or total [tex]120 \times 21 = 2520[/tex]

Thus,

The number of seating arrangements possible in this case is 2520

Learn more about seating arrangements here:

https://brainly.com/question/13605688