Probability: arrangements with identical elements

kingwinner

1) If we have 5 letters: A, A, B, C, D, in how many ways (arrangements) can we form a 3-letter "word"?

How can I calculate this? The 2 A's seem to make things very complicated...and I have no clue how to do it...

It's arrangements, so I think permutation will be used.

Also, there are two identical "A"s, so some arrangements will be double counted or so...

Denote 1st A=A1 and 2nd A=A2
A1BC and A2BC are counted as 1 "word" because actually A1=A2

Thanks for helping!

HallsofIvy

Yes, that is a little complicated! If you do as you say, and label the As as A1 and A2, there 5 distinct letters and so 5!/3!= 20 distinct words. However, you are correct that that is over counting because, in fact, some words are "the same". Try this: how many words are there that contain no As? That's easy- with no As, you just have BCD and there are 3! of them (the different permutations of BCD). How many different three letter words are there if you include only one A? That's again easy because now you are just looking at {A, B, C, D} with only one A. Finally, how many three letter words are there if you include both As? AAB, ABA, BAA, AAC, ACA, CAA, AAD. ADA, DAA: that's all the possibliities and there are exactly 9 of them: the number of three letter distinct words that can be formed from A, A, B, C, D is the sum of those three numbers. (It is surprizingly close to 20!)

kingwinner

"If you do as you say, and label the As as A1 and A2, there 5 distinct letters and so 5!/3!= 20 distinct words" <---shouldn't it be 5 x 4 x 3 = 60 words in this case?


So is the answer to #1 going to be 3! + (4 x 3 x 2) + the number of possible arrangments with exactly 2 A's ?
Also, is there any way to find the number of possible arrangements with exactly 2 A's without having to LIST them all out?


Thanks!

learningphysics

careful 4*3*2 isn't the number of arrangements with exactly 1 A... you're including possibilities where A isn't picked.

kingwinner

Oh no...then how can I find the number of arrangements with exactly 1 A? I am so confused...

learningphysics

You have 3 positions for the 1A... so that's 3 possibilities... for each of those possibilities, you have 2 slots to put in B,C or D...

so you 3*P(3,2)

or in other words 3*(3*2)

Sleek

Sorry, I'll have to reconsider my solution.

kingwinner

Thanks, I understand this part now!

And my final question is:
Is there any way to calculate the number of possible arrangements correctly with exactly 2 A's without having to LIST them all out? (because it's very easy to miss something by listing, in my opinion)

learningphysics

Yes, it's very similar to analyzing the 1 A case. For 2A's... you have 3 possible positions for the 3rd letter. And for each of those positions, you can make 3 choices for the 3rd letter... B, C or D. So you have 3*C(3,1) = 3*3 = 9

kingwinner

Nice, now I understand it perfectly. Thank you!

kingwinner

Combinations with identical elements:
2) You have 2 blue balls, 3 yellow balls, and 3 red balls in a box. If you randomly take 3 balls from the box, what is the probability that you have 1 ball of each color? (assume order does not matter)

This is even crazier, how can I even get a starting point for this problem? I know the definition of combination, but I still have absolutely no clue on solving this problem...

Any help is appreciated!

learningphysics

Give it a shot yourself. Make a guess... we'll help you along.

kingwinner

My guess:
2C1 x 3C1 x 3C1
--------------------
(8 x 7 x 6) / (2! 3! 3!)

But I don't think it's right...

learningphysics

I think you're actually quite close... your numerator is correct. it is the number of ways to choose a blue ball, yellow ball and a red ball.

So all you need in the denominator is the total number of ways of choosing 3 balls out of 8.

kingwinner

2) So would it be
2C1 x 3C1 x 3C1
---------------- ?
8C3

Would this in the denominator account for the identical balls?



Also, do I have to multiply the numerator by 6? (since there are 5 mutually exclusive events that can give one ball of each color when order doesn't matter.

The 6 mutually exclusive events are:
RBY (first ball red, second ball blue, third ball yellow)
RYB
BYR
BRY
YBR
YRB


Thanks!

learningphysics

Yes, this would work.

We aren't using permutations in the numerator or denominator. We're using combinations. If we were using permutations, then in the numerator we'd have 2C1 x 3C1 x 3C1 x 3! and in the denominator we'd have P(8,3). This would also work... it gives the same answer.

What wouldn't work is having permutations in the numerator and cominbations in the denominator... or combinations in the numerator and permutations in the denominator.

Using permutations in both the numerator and denominator... or combinations in both the numerator and denominator... will work.

kingwinner

Thanks! I am sort of getting it now.


But what if I change the question to:
You have 2 blue balls, 3 yellow balls, and 3 red balls in a box. If you randomly take 3 balls from the box, what is the probability that you have the first ball red, second ball yellow, and third ball blue? (order DOES matter)