# Counting combinations

## Homework Statement

In how many ways can 4 articles be divided between 2 people when each person must receive at least one article?

## The Attempt at a Solution

I tried it like this

Person 1 could have ( 3,2 or 1 paper(s) )

Person 2 the same so

total # ways = 6 + 6 = 12

But this is wrong.

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Simon Bridge
Homework Helper
The number of articles each person has is not independent.
It is not clear that the articles are identical - suspect not.

Label the articles 1,2,3,4
(note: an "article" need not be a paper - it can be any object, like "an article of clothing".)

Label people as A and B (Alice and Bob, say).

Now list the different ways Alice and Bob can have some stuff.
Start by giving Alice one article and Bob the rest.
Then give Alice two articles, then three.

So like A-1, B-3 = 3 ways
A-2, B-2 = 4 ways

A-3,B-1 = 3 ways
So total # = 10 ways?

Nevermind I got it thanks.

Simon Bridge
Homework Helper
Well done: what did you come up with?
(JIC someone else gets stuck - then they benefit from your efforts)

I had the first possibility ( 4C1 x 3C3)
2nd (4C2 x 2C2)
3rd (4C1 x 3C3)

Then I added them up to get the total number of ways of carrying out the selections. I got 14.

Simon Bridge
Homework Helper
Cool - what made you suddenly switch to combinations notation?

The question amounts to asking the number of ways Alice can pick at most 3 articles out of 4 when the order doesn't matter.

That's (4x3x2)/(3x2)+(4x3)/(2)+4=4+6+4=14.

There's so few you can just list them.

Usually when I can't count it out lol I try Combinations, some times it just seems easier with combinations.