Question on permutation and combination(girls and boys standing in a line)

In summary, the conversation is discussing a problem involving positioning a block of boys and girls in a line and shuffling them around. The question asks for the number of ways this can be done, which is determined to be 5! x 3! x 4C1, or 4 x 5! x 3!. The discussion also touches on whether this is a permutation or combination problem, with some disagreement on the answer. It is ultimately concluded that it is a permutation problem, but also acknowledges that there may be some elements of both involved.
  • #1
mutineer123
93
0

Homework Statement

http://www.xtremepapers.com/CIE/International%20A%20And%20AS%20Level/9709%20-%20Mathematics/9709_w03_qp_6.pdf
question 6 b ii

Homework Equations


The Attempt at a Solution


The concept is bothering me. the answer is 5! × 3! × 4C1. Why is it a permutation problem? Plus why do we use 5! × 3! in a permutation problem, isn't it used in combinations? as 5!X 3! shows the nos of ways it can be arranged 'in any order' So how can we use that in a permu where order DOES matter!

The reason I think its a permutation problem is because of http://answers.yahoo.com/question/index?qid=20110530100750AAXrtRp
other than that I don't relly know why it is a permutation problem even.
 
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  • #2
Draw a picture. How many ways can you position a block of 5 boys in a line of 8 people with 3 girls? Then given that number of patterns how many ways can you shuffle the boys and girls around in their positions?
 
  • #3
Dick said:
Draw a picture. How many ways can you position a block of 5 boys in a line of 8 people with 3 girls? Then given that number of patterns how many ways can you shuffle the boys and girls around in their positions?
I don't think you got what I was trying to say. Yes i know 'how' it is 5! X 3! ways, but i don't know 'why' it is a permutation, because they are clearly being selected randomly...
 
  • #4
mutineer123 said:
I don't think you got what I was trying to say. Yes i know 'how' it is 5! X 3! ways, but i don't know 'why' it is a permutation, because they are clearly being selected randomly...

Ok, so you know the solution. If it's any help, I would not have written 4C1 instead of 4. The answer is just 4*5!*3!. Why you would write 4C1 instead of 4 escapes me. Is that what you are asking?
 
  • #5
Dick said:
Ok, so you know the solution. If it's any help, I would not have written 4C1 instead of 4. The answer is just 4*5!*3!. Why you would write 4C1 instead of 4 escapes me. Is that what you are asking?

No, ok firstly, do you agree its a permutation sum and NOT a combination?
 
  • #6
mutineer123 said:
No, ok firstly, do you agree its a permutation sum and NOT a combination?

I agree that it has somewhat more to do with permutations, but I also don't think there is an absolutely clear separation between permutation problems and combination problems. Sometimes you have to use a little of both.
 
  • #7
I hesitate to do this since I'm not well-versed in combinations/permutations, but don't we have more than 4(5!)(3!)? The 5! comes from the block of boys, and the 3! from a block of girls, but aren't there more different combinations of girls around the boys? 1 girl in front, 2 in back, 2 in front 1 in back, 3 in front 0 in back. I'm wondering myself how those are accounted for (not to hijack the thread...).
 
  • #9
zooxanthellae said:
I hesitate to do this since I'm not well-versed in combinations/permutations, but don't we have more than 4(5!)(3!)? The 5! comes from the block of boys, and the 3! from a block of girls, but aren't there more different combinations of girls around the boys? 1 girl in front, 2 in back, 2 in front 1 in back, 3 in front 0 in back. I'm wondering myself how those are accounted for (not to hijack the thread...).

You forget another one, 3 in the back 0 in front.
<--
GBGG or 3!5!
GGBG or 3!5!
GGGB or 3!5!
BGGG or 3!5!
=4.3!5!

Anymore combination you can add?
 
Last edited:
  • #10
azizlwl said:
You forget another one, 3 in the back 0 in front.
<--
GBGG or 3!5!
GGBG or 3!5!
GGGB or 3!5!
BGGG or 3!5!
=4.3!5!

Anymore combination you can add?

Ah, got it. Thanks.
 

FAQ: Question on permutation and combination(girls and boys standing in a line)

What is the difference between permutation and combination?

Permutation refers to the arrangement of a set of objects in a specific order, while combination refers to the selection of a subset of objects from a larger set without any particular order. In the context of girls and boys standing in a line, permutation would refer to the different ways in which they can be arranged in a line, while combination would refer to the different ways in which a specific number of girls and boys can be selected from the group.

How do you calculate the total number of possible arrangements for a given number of girls and boys standing in a line?

To calculate the total number of possible arrangements, we can use the formula for permutation, which is n!/(n-r)!, where n is the total number of objects and r is the number of objects being arranged. In this case, the total number of objects would be the number of girls plus the number of boys.

Can you give an example of a question about permutation and combination involving girls and boys standing in a line?

One example could be: In a group of 5 girls and 3 boys, how many different ways can they stand in a line if the girls and boys must alternate? In this case, we would use the formula for permutation, as the order in which they stand matters.

How does the number of possible arrangements change if some of the girls or boys are identical?

If some of the girls or boys are identical, the number of possible arrangements decreases. This is because having identical objects reduces the number of distinct arrangements. In this case, we would use the formula for combination, as the order does not matter.

Are there any real-life applications of permutation and combination involving girls and boys standing in a line?

Yes, there are several real-life applications. For example, in a school assembly, students may be asked to stand in a line with girls and boys alternating. In a sports team, players may be lined up in a particular order with girls and boys alternating. In a choir performance, singers may be arranged in a line with girls and boys alternating. In each of these scenarios, the concept of permutation and combination would be applicable.

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