Math Problem....

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• 11-02-2002
NANO
Math Problem....
Hie everybody....

This Problem drive me very crasy;

before I will write the problem, I will tell you the answer is not as easy as I thought... it needs a lot of writing to get the right answer.

and this is the problem
Code:

```  problem How many ways are there to distribute five identical penciles and eight different notebooks amonge five poor math students,  in case that each student getting at least one pencil and one notebook?```
:o
• 11-02-2002
crag2804
32768
• 11-02-2002
Magos
Quote:

problem

How many ways are there to distribute five identical penciles and
eight different notebooks
amonge five poor math students,
in case that each student getting at least
one pencil and one notebook?
Hmm, it was some time ago I read discrete mathematics... but I'll try :).

The 5 pencils can only be distributed in 1 way since they are identical, and have the same amount as the students.

The placement of the eight books can be split into 3 parts, as shown in the figure.

In part 1, the books can be distributed in 8! ways. But since the order of the books isn't important in column 5, you must divide by 4! so you get 8! / 4!. Also note that the fifth column can be in any of the five students, so multiply that by 5.
You get (8! / 4!) * 5.

In part 2, the books can be distributed in 8! ways. But the order of the books isn't important in the two last columns so you have to divide by 2! and 3!. You get 8! / (2! * 3!). The last two columns can be in any of the five students, so you have to multiply by 5 * 4 (the last column can be in any of the 5, the second last can be in any of the 5 minus the one the last one is in, giving 4).
You get (8! / (2! * 3!)) * (5 * 4).

In part 3, the books can be distributed in 8! ways. But the order of the books isn't important in the three last columns, so you have to divide by 2! three times. You get (8! / (2! * 2! * 2!)). The last three columns can be in any of the five students, so you have to multiply by 5 * 4 * 3.
You get (8! / (2! * 2! * 2!)) * (5 * 4 * 3).

Then sum all those:

(8! / 4!) * 5 +
(8! / (2! * 3!)) * (5 * 4) +
(8! / (2! * 2! * 2!)) * (5 * 4 * 3)

You might want to shorten that expression:
8! * ((5 / 24) + (10 / 6) + (15 / 2))

And the final result:
378000

There :).
• 11-02-2002
Magos
Oops, forgot to attach the image :o
• 11-02-2002
Sang-drax
I'd like to try:

The pencils don't have to be considered.

First, give every student one book each; this can be done in 8*7*6*5*4 = 6720 ways

Then, distribute the remaining three books. Ways to do this: 5*5*5 = 125

Total number of distributions : 6720 * 125 = 840000

Not the same answer as Magos :(

***

Lets try to simplify the problem; with three books and two students , the answer is 12 (use pen and paper).

My algorithm gives:
*Every student one book : 3*2 = 6 ways
*Distribute the remaining 1 books : 2 ways

Magos' algorithm gives:
(only one part to consider)
*3! ways, but order of the books in column 2 isn't important, so divide with 2!
*answer is 3!/2! = 3, wrong

Is my solution better, or even correct? I'm not sure, I can't find the flaw in neither mine nor Magos' algorithm.

(Edit) This is not correct... :( (/Edit)
• 11-02-2002
crag2804
How many different combinations of letters are there in a one letter word?

26 e+1 = 26

two letter word = 26 e +2 = 676
...
...

With 8 different books and only 5 students shouldn't that just therefore be
8 e+5 = 32768

I don't know, just a stab in the dark. If I'm wrong can someone explain why.

I figured out a hole in my method, it doesn't take into account that fact that the books can't be used more than once.
Shouldn't that therefore mean that there are less than 32768 though?
Which would give a figure different to everyones
[/edit]
• 11-02-2002
Cshot
>> Total number of distributions : 6720 * 125 = 840000

EDIT - I think you're right.

EDIT#2 - Hehe, made me edit again. I knew my original reasoning was correct :)
Maybe I should hit those math books again...nah
• 11-02-2002
Nick
I havn't done a problem like this in a long time, but here goes.

Because each of the pencils is indentical they may effectivly
be eliminated from the problem. Therefore
8 * 7 * 6 * 5 * 4 * (5 choose 3)
This follows because for the first student there are 8 books
the next student 7 books to choose from. The final 3 aren't constrained so we have 5 choose 3.
This equals 67200.
• 11-02-2002
Nick
Sang Dang's answer is correct. I solved the problem without
• 11-02-2002
PJYelton
Sang-drax, there should only be 6 different ways to distribute 3 books to 2 students. Say you have a red, green, and blue notebook then the combos are:

R GB
GB R
G BR
BR G
B RG
RG B

Your method counts "R GB" as being different than "R BG"
• 11-03-2002
Magos
Quote:

Originally posted by Sang-drax
Lets try to simplify the problem; with three books and two students , the answer is 12 (use pen and paper).

My algorithm gives:
*Every student one book : 3*2 = 6 ways
*Distribute the remaining 1 books : 2 ways

Magos' algorithm gives:
(only one part to consider)
*3! ways, but order of the books in column 2 isn't important, so divide with 2!
*answer is 3!/2! = 3, wrong

Is my solution better, or even correct? I'm not sure, I can't find the flaw in neither mine nor Magos' algorithm.

Sorry, but you can only distribute 3 books to 2 students in 6 ways, as shown in the picture below. And my algorithm gives that answer, not yours. You forgot to multiply by 2, since the "two-book-column" can be in any of the two students.
(3! / 2!) * 2 = 3! = 6

Quote:

Sang Dang's answer is correct. I solved the problem without

No it's not, as shown above!
• 11-03-2002
Magos
Quote:

Originally posted by crag2804
How many different combinations of letters are there in a one letter word?

26 e+1 = 26

two letter word = 26 e +2 = 676
...
...

With 8 different books and only 5 students shouldn't that just therefore be
8 e+5 = 32768

I don't know, just a stab in the dark. If I'm wrong can someone explain why.

I figured out a hole in my method, it doesn't take into account that fact that the books can't be used more than once.
Shouldn't that therefore mean that there are less than 32768 though?
Which would give a figure different to everyones
[/edit]

When you try to solve this problem using combinations of letters, you don't take in account that:

A B | C

Isn't the same as:

A | B C

(| separates the two columns)
• 11-03-2002
Magos
Quote:

Originally posted by Nick
Therefore 8 * 7 * 6 * 5 * 4 * (5 choose 3)
This follows because for the first student there are 8 books
the next student 7 books to choose from. The final 3 aren't constrained so we have 5 choose 3.
This equals 67200.

If we take your algorithm and use it in the case with 2 students and three books (which we knows the answer of), you get:

3 * 2 * (2 choose 1) =
3 * 2 * (2! / (1! * (2 - 1)!)) =
3 * 2 * 2! =
3 * 2 * 2 =
12