Divinding 8 teachers up among 4 schools.

28. If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?

OK, so it's pretty clear that if is the number of teachers allocated to one of the four schools then

Comparing this too the chapter text leads me to the following, , which is the number of distinct nonnegative integer-valued vectors satisfying the equation

So my first question is, since I need , to be a valid entry does this formula cover that case? (I'm thinking it does because it mentions nonnegative integer-valued vectors.)

Secondly, taking and , I've got as my current working solution. However, the text is talking about indistinguishable objects here and I hardly think teachers are indistinguishable. Also, the listed answer is 65,536. So I think I need to permute the teachers now somehow? Or is this all wrong?

Re: Divinding 8 teachers up among 4 schools.

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**bkbowser** 28. If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?

OK, so it's pretty clear that if

is the number of teachers allocated to one of the four schools then

Comparing this too the chapter text leads me to the following,

, which is the number of distinct nonnegative integer-valued vectors

satisfying the equation

So my first question is, since I need

, to be a valid entry does this formula cover that case? (I'm thinking it does because it mentions nonnegative integer-valued vectors.)

Secondly, taking

and

, I've got

as my current working solution. However, the text is talking about indistinguishable objects here and I hardly think teachers are indistinguishable. Also, the listed answer is 65,536. So I think I need to permute the teachers now somehow? Or is this all wrong?

for your first question the simple example of n=1, r=2 shows that the case of is covered by this formula.

for your second question, ignoring what you've posted, think about it this way. Each teacher can be assigned to 1 of 4 schools. Think of it as giving them a base 4 digit. there are 8 teachers so the total number of variations is .

Re: Divinding 8 teachers up among 4 schools.

Re: Divinding 8 teachers up among 4 schools.

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**romsek** for your first question the simple example of n=1, r=2 shows that the case of

is covered by this formula.

for your second question, ignoring what you've posted, think about it this way. Each teacher can be assigned to 1 of 4 schools. Think of it as giving them a base 4 digit. there are 8 teachers so the total number of variations is

.

With zero would have to be an option since we only care about integer solutions. . OK good.

Assigning each teacher a number out of a base 4 system seems to work well, but can I use the

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Originally Posted by

**Plato** First I assume that there are two distinct questions here. 1) there are no restrictions. 2) each school gets two teachers.

1) There are

functions from a set of

elements to a set of

elements.

2) How many ways can we rearrange the string

Sorry, yes there are two questions in the original, I didn't actually get around to trying to solve the second as I wasn't finished with the first.

1) looks very similar to romsek's suggestion. Is there no way to use this proposition from the text to solve part 1? , which is the number of distinct nonnegative integer-valued vectors satisfying the equation .

For 2) there are essentially 8 slots and 8 options for the first slot. The options decrease by one with every slot so there are 8! ways? No this can't be right.

Re: Divinding 8 teachers up among 4 schools.

Quote:

Originally Posted by

**bkbowser** 1) looks very similar to romsek's suggestion. Is there no way to use this proposition from the text to solve part 1?

, which is the number of distinct nonnegative integer-valued vectors

satisfying the equation

.

For 2) there are essentially 8 slots and 8 options for the first slot. The options decrease by one with every slot so there are 8! ways? No this can't be right.

NO! 1) is . The number of functions from eight teachers to four schools.

2) the answer is .

Think of the schools as Make a list of the eight teachers.

Arrange the string , one next to each name.

Every rearrangement of that string is a possible way to assign eight teachers to four schools with two to each school.

Re: Divinding 8 teachers up among 4 schools.

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**Plato** NO! 1) is

. The number of functions from eight teachers to four schools.

2) the answer is

.

Think of the schools as

Make a list of the eight teachers.

Arrange the string

, one next to each name.

Every rearrangement of that string is a possible way to assign eight teachers to four schools with two to each school.

I don't fully follow but I think I can make sense of it with the text as follows; So this is a Multinomial Coefficient problem where each of the distinct subgroups is 2, which the proof in the text shows is equal to , and this is equal to .

Does this look good?

Re: Divinding 8 teachers up among 4 schools.

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**bkbowser** and this is equal to

.

Does this look good?

Yes, that is correct.

But the other model is very useful.

Do you understand how many ways you can rearrange the word

Well .

That is also how many ways that a collection of eleven people can be divided into four different cells where two cells contain four people, another contains two and the fourth contains one.

Re: Divinding 8 teachers up among 4 schools.

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**Plato** Do you understand how many ways you can rearrange the word

Well

.

That is also how many ways that a collection of eleven people can be divided into four different cells where two cells contain four people, another contains two and the fourth contains one.

I'm not sure I understand the model.

11! is the permutations of an 11 place word.

I know you have to take out the number of duplicates, which I presume is the denominator, I'm just not sure how.

Re: Divinding 8 teachers up among 4 schools.