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# Counting questions

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A. In how many ways can five red chairs and three white chairs be arranged such that no two white chairs are next to each other?

B. In how many ways can four red chairs, three white chairs, and two blue chairs be arranged such that there is at least one red and one blue chair between two white chairs?

For A, there are initially 8!/5!3! = 5040 ways to arrange the chairs. Using complementary counting will be easier, so I just counted the number of ways that at least two white chairs are next to each other. There are 7 ways for 2 white chairs to be next to each other and 6 ways for 3 white chairs to be next to each other. Subtract this from 5040 to get 5027. Could somebody check this?

As for B, I'm not sure how to count the number of ways.

thank you in advance!

Jun 8, 2022

#1
+125695
+2

Here's what I get

A)   Assuming that the chairs  of each color are identical

You have calculated the  the number of possible arrangements correctly

If two white chairs  are together  they could occupy  positions

1,2    and the other white chair can occupy positions  4,5,6,7 or 8

2,3    and the other white chair can occupy positions 5,6,7 or 8

3,4    and the other white chair can occupy positions 1, 6, 7 or 8

4,5    and the other white chair can occupy  positions 1,2, 7 or 8

5,6   and  the other white chair can occupy  positions  1,2,3 or 8

6,7   and the othe white chair can occupy  positions  1,2,3 or 4

7,8   and the other white chair can occupy positions 1,2,3,4,5

So.....we have   30  arrangements here

Three white chairs together  can occupy   any of 6 positions

So......the total number of excluded arragements =     36

So....the total number of  acceptable arrangememts =   5040   -  36   =   5004

Jun 8, 2022
#2
+118572
+1

Here is my take.

A

In how many ways can five red chairs and three white chairs be arranged such that no two white chairs are next to each other?

I am also assumiong that all the white chairs are identical and all the red chairs are identical.

And I am assuming that the chairs are placed in a row

The white chairs must be seperated so I can start with

W R W R W

Now there are 3 more red ones so I will put hashes  where one or more of them can go

# W # R W # R W #

If all three are together then there are 4 places they can go

If 2 are together and the other seperate then there are 3+3+3+3=12 ways

If none are placed together then we have  4C3=4ways

4+12+4= 20 ways

20 ways is my answer

B

B. In how many ways can four red chairs, three white chairs, and two blue chairs be arranged such that there is at least one red and one blue chair between two white chairs?

The white chairs are seperated by at lease 1 red and one blue chair so there is the start

W [BR] W [BR] W

Of course the BR  could also be RB and that goes for the second lot of BR as well so there are 4 copies of all continuing arrangements

(I could be double overcounting but I don't think so)

So

W [BR] W [BR] W      Ive used all the White and all the blue chairs already there are only 2 red ones unused so where can I put them?

# W #[BR]# W #[BR]# W #

If they both stay together then there are  6 places they can go

If they are seperated then there are  6C2 = 15 places they can go.

Which adds to 15+6=21 places

But there were 3 other starter arrangments

4*21 = 84 possible arrangments

Jun 9, 2022
#3
+118572
0

We will expect a response from you grs75.

Preferably with the answer you are given as correct.

So far you do not have flawless history of showing appreciation.

https://web2.0calc.com/questions/question-on-polynomial-division-question#r2

Melody  Jun 9, 2022
#4
+88
+2

I'm sorry. I was expecting to receive an email notification but did not get one, and I am not on this website 24/7. I did not intend to come off as unappreciative.

As for the correct answers, the answer to part A is 20. A pair of white chairs must be separated by one red chair. The red chairs can be arranged like _ R _ R _ R _ R _ R _ with 6 spaces to put the white chairs. Then 6 choose 3 = 20. Part B is not so clear to me, so I'm not sure of the correct answer.

grs75  Jun 10, 2022
edited by grs75  Jun 10, 2022
edited by grs75  Jun 10, 2022
#5
+118572
+1

Thanks Grs75.

Your response is appreciated.

It seems my part A and your given answer are the same anyway.

That is a good start :)

Melody  Jun 10, 2022
#6
+88
+1

The answer to B is apparently 60. I cannot understand why though.

grs75  Jun 10, 2022
#7
+118572
+3

ok thanks for that, I thought I might have double counted.

Ok here is the go

the first bit of what I did was correct

# W #[BR]# W #[BR]# W #

If they both stay together then there are  6 places they can go

If they are seperated then there are  6C2 = 15 places they can go.

Which adds to 15+6=21 places

now lets look at a scenario

consider starting with

# W #[BR]# W #[RB]# W #

If I added 2 reds tied together they will all be different than any permutation in the first set, that is 6 ways

If 2 two added reds are to be added seperately then

# W #[BR]# W R[BR]# W #

will all be the same as

# W #[BR]# W #[RB]R W #

and there are 4 places where the second R can be put, so if I add15 in this second set I will have double counted by 4

So rather than 15 there will be 11

The next set will have 4 less again and the same with the last set

So rather than it being

21+21+21+21

it will be

21+17+13+9 = 60

Melody  Jun 11, 2022
#8
+88
+1

I think i get it, thank you!

grs75  Jun 12, 2022
#9
+118572
+1

You are welcome.

Melody  Jun 13, 2022
#10
+2665
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Here's another way for 1 using Complementary Counting:

There are $${8 \choose 5} = 56$$ ways to order them.

If only 2 white chairs are together, there are 7 positions they can occupy (1 and 2, 2 and 3, 3 and 4, and so on)

If the white chairs occupy spots (1 and 2) or (7 and 8), there are 5 options on how to order the remaining chairs, making for 10 ways (the white chair can't be directly next to the white chairs).

For the remaining 5 spots, there are only 4 ways to put the remaining white chair, making for 20 ways.

This means that there are $$5 \times 4 + 5 + 5 = 30$$ ways if only 2 white chairs are next to each other.

Next, there are only 6 ways for all 3 white chairs to be next to each other (1,2, and 3; 2, 3, and 4; and so on).

This makes for $$56 - 30 - 6 = \color{brown}\boxed{20}$$ ways.

Jun 13, 2022