Question

A queue has 4 boys and 4 girls standing in line. Find how many different arrangements of the line there are possible if the boys and girls alternate? Can someone please teach me how to get the answer thanks :)

Answer 1

There are 8 positions in the queue.
There are 4 choices of a particular boy for the lst "boy spot," 3 choices for the 2nd "boy spot" and continuing likewise to one choice for the 4th "boy spot."

So, the boys can be arranged in 4! ways where 4! means 4 times 3 times 2 times 1.

The girls can be also be arranged in 4! ways using the same reasoning as that for the boys.

Because we do not know if a boy or if a girl heads up the line, there are 2 ways to arrange the queue: one with boys in the odd numbered spots and girls in the even numbered spots. The second is with girls in the odd numbered spots and boys in the odd spots.

So, the line can be arranged in the following number of ways:

4!*4!*2.

Answer 2

B G B G B G B G or G B G B G B G B
4*4*3*3*2*2*1*1 or 4*4*3*3*2*2*1*1

4! 4! 2

Answer 3

how many arrangements would there be if 2 particular girls wish to stand together? 4!3!4!?

Answer 4

Did you understand how the first part is done?

Answer 5

yes

Answer 6

Try your hand at the second part.

Answer 7

Oh, I see that you did. >> 4!3!4!?
What is your reasoning on that?

Answer 8

well it doesnt work because i just put it into the calculator and it comes up with syntax error

Answer 9

i thought that there are 4! ways for the first girl and there would be 3! ways for the rest of the girls and 4! ways for any boys?

Answer 10

Regarding syntax error, use the Google calculator.

Answer 11

its not right because the answer book has 10080 as the answer

Answer 12

Are the boys and girls still alternating in the line on part b? Or, was that condition only for part a?

Answer 13

no they're not alternating its a different question but with the same # of boys and girls in a queue

Answer 14

ok

Answer 15

Consider the two girls to be bound together, one in front of the other.
That leaves us with 7 positions for the line with one of the positions having two people.

7*6*5*4*3*2*1 ways BUT the two girls standing in one of the seven slots can be arranged two ways in that slot.

So, the total number of ways is [ 2 * 7*6*5*4*3*2*1 ] which equals ?

Answer 16

Let me know that equals. Okay?

Answer 17

10080

Answer 18

That is what you said is correct, yes?

Answer 19

i dont understand how you get that?

Answer 20

its correct

Answer 21

>i dont understand how you get that?


Did you see the explanation?

Consider the two girls to be bound together, one in front of the other.
That leaves us with 7 positions for the line with one of the positions having two people.

7*6*5*4*3*2*1 ways BUT the two girls standing in one of the seven slots can be arranged two ways in that slot.

So, the total number of ways is [ 2 * 7*6*5*4*3*2*1 ] which equals ?

Answer 22

how do you know its only 2 ways though ?

Answer 23

In the problem statement, this was given: 2 particular girls wish to stand together. That is the origin of the two.

Answer 24

The answer is not two. The answer is 2 * 7*6*5*4*3*2*1 which is 10,080.