Math Help - Boys and girls sitting in a row.

1. Boys and girls sitting in a row.

These problems are all sub problems of number 7, I've been assuming that they are all related, and there are no additional instructions not listed. I'm mostly just looking for confirmation that my solution is entailed from the premises.

7(b) In how many ways can 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together?

I'm assuming the first slot, of six total, can be filled by anyone so there are six possibilities.

The second slot has to be filled by someone of the opposite sex so there are three possibilities.

The third and fourth slots work the same as the first two slots only there are now 4 total, and 2 of the opposite gender to select from.

Similarly for the last two slots.

So I'm claiming that this works out to be $(6*3)(4*2)(2*1)=288$

Since the given answer is 72 this must be wrong. And I haven't been able to work out a way that ends with the correct answer. Like I've tried a few different interpretations of the question and I still get nothing.

Also, since 288 is an integer multiple of 72, there must be a factor of 4 in my solution someplace it isn't supposed to be.

7(c) In how many ways if only the boys must sit together?

So I'd guess this works out too;

bbb,ggg
g,bbb,gg
gg,bbb,g
ggg,bbb

where any given boy or girl can be any of the three, so;

(3*2*1)(3!)
3*(3!)*(2!)
3*2*(3!)*1
3!*3!

or just $(3!*3!)*4=144$ which is listed as the correct answer. Is this solution correct?

7(d) In how many ways if no two people of the same sex are allowed to sit together?

This one seems strange; is boy, boy, boy a good configuration? As no two people of the same sex are sitting together.

b,g,b,g,b,g which is $3*3*2*2*1*1=36$

and alternately

g,b,g,b,g,b

For a total of 72 combinations, which is the listed answer. Is this solution correct?

2. Re: Boys and girls sitting in a row.

Originally Posted by bkbowser
7(b) In how many ways can 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together?
There are $3!=6$ ways that the boys to sit together and also the girls to sit together.
Thus $(6)(6)(2)=72$. Where does that two come from?

Originally Posted by bkbowser
7(c) In how many ways if only the boys must sit together?
$(6)(4!)=144.$

Originally Posted by bkbowser
7(d) In how many ways if no two people of the same sex are allowed to sit together?
$(3!)(3!)(2)$ seat a boy in every other chair then seat the girls. Again why the 2?

3. Re: Boys and girls sitting in a row.

Originally Posted by Plato
There are $3!=6$ ways that the boys to sit together and also the girls to sit together.
Thus $(6)(6)(2)=72$. Where does that two come from?
I'm not sure which two you are referring too and I can't explain how I got to that conclusion any better then I have already.

It looks like your configuration is;

bbb,ggg or ggg,bbb

in either case the permutations would be 3!*3!

then you take both cases together to make, 3!*3!*2?

Originally Posted by Plato
$(6)(4!)=144.$
I don't see how you could arrive at that. Where does the 6 come from? Where does the 4! come from?

Originally Posted by Plato
$(3!)(3!)(2)$ seat a boy in every other chair then seat the girls. Again why the 2?
What two? Aside from some syntax our solutions look identical.

4. Re: Boys and girls sitting in a row.

Originally Posted by bkbowser
I don't see how you could arrive at that. Where does the 6 come from? Where does the 4! come from?
Think of the boys as a single block. There are six ways to arrange the boys within that block.
That block along with the three girls make four individuals to seat in the row. Six people in all and the boys together as a block.

5. Re: Boys and girls sitting in a row.

bkbowser: your explanations for 7(c) and 7(d) are perfectly fine - Plato is simply showing different noemnclature.

As for 7(b) - you seem to have interpreted the meaning of "the boys and the girls are each to sit together" differently than either Plato or the book. Your solution is what you get if each pair of seats must have a boy and girl, so for example you would say this is allowable:

gbgbbg

whereas I think what they're looking for is how many ways can all 3 girls sit side-by-side and all 3 boys also sit side-by-side. Thus the allowable configurations are gggbbb and bbbggg. For each of these arrangements there are 3! ways to seat 3 girls is row, and 3! ways to seat 3 boys in a row, so the total number of ways is 2 x 3! x 3! = 72.

6. Re: Boys and girls sitting in a row.

Originally Posted by Plato
Think of the boys as a single block. There are six ways to arrange the boys within that block.
That block along with the three girls make four individuals to seat in the row. Six people in all and the boys together as a block.
Originally Posted by ebaines
bkbowser: your explanations for 7(c) and 7(d) are perfectly fine - Plato is simply showing different noemnclature.

As for 7(b) - you seem to have interpreted the meaning of "the boys and the girls are each to sit together" differently than either Plato or the book. Your solution is what you get if each pair of seats must have a boy and girl, so for example you would say this is allowable:

gbgbbg

whereas I think what they're looking for is how many ways can all 3 girls sit side-by-side and all 3 boys also sit side-by-side. Thus the allowable configurations are gggbbb and bbbggg. For each of these arrangements there are 3! ways to seat 3 girls is row, and 3! ways to seat 3 boys in a row, so the total number of ways is 2 x 3! x 3! = 72.
OK, that makes perfect sense. Thanks you two.