Probability: arrangements with identical elements

In summary: 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
  • #1
kingwinner
1,270
0
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!
 
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  • #2
kingwinner said:
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!
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!)
 
  • #3
"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!
 
  • #4
kingwinner said:
"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!

careful 4*3*2 isn't the number of arrangements with exactly 1 A... you're including possibilities where A isn't picked.
 
  • #5
learningphysics said:
careful 4*3*2 isn't the number of arrangements with exactly 1 A... you're including possibilities where A isn't picked.

Oh no...then how can I find the number of arrangements with exactly 1 A? I am so confused...
 
  • #6
kingwinner said:
Oh no...then how can I find the number of arrangements with exactly 1 A? I am so confused...

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)
 
  • #7
Sorry, I'll have to reconsider my solution.
 
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  • #8
learningphysics said:
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)

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)
 
  • #9
kingwinner said:
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)

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
 
  • #10
learningphysics said:
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

Nice, now I understand it perfectly. Thank you!
 
  • #11
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!
 
  • #12
kingwinner said:
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!

Give it a shot yourself. Make a guess... we'll help you along.
 
  • #13
learningphysics said:
Give it a shot yourself. Make a guess... we'll help you along.

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

But I don't think it's right...
 
  • #14
kingwinner said:
My guess:
2C1 x 3C1 x 3C1
--------------------
(8 x 7 x 6) / (2! 3! 3!)

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

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.
 
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  • #15
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!
 
  • #16
kingwinner said:
2) So would it be
2C1 x 3C1 x 3C1
---------------- ?
8C3

Would this in the denominator account for the identical balls?

Yes, this would work.

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!

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.
 
  • #17
learningphysics said:
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.
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)
 
  • #18
kingwinner said:
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)

Then it would be (2*3*3)/P(8,3)
 
  • #19
learningphysics said:
Then it would be (2*3*3)/P(8,3)

Why?

Why not
2P1 x 3P1 x 3P1
---------------- ?
8P3
 
  • #20
kingwinner said:
Why?

Why not
2P1 x 3P1 x 3P1
---------------- ?
8P3

Same thing. 2P1 = 2. 3P1 = 3.
 

1. What is the formula for calculating the number of arrangements with identical elements?

The formula for calculating the number of arrangements with identical elements is n!/a!b!c!, where n is the total number of elements and a, b, and c represent the number of identical elements of each type.

2. How is the formula for arrangements with identical elements different from the formula for permutations?

The formula for arrangements with identical elements takes into account the number of identical elements, while the formula for permutations does not. This means that the formula for arrangements will result in a smaller number of possible arrangements.

3. Can the formula for arrangements with identical elements be used for combinations?

No, the formula for arrangements with identical elements cannot be used for combinations because combinations do not take into account the order of the elements. The formula for combinations is n!/(r!(n-r)!), where n is the total number of elements and r is the number of elements being selected.

4. How can the formula for arrangements with identical elements be applied in real-world scenarios?

The formula for arrangements with identical elements can be applied in situations where there are a certain number of distinct objects that can be arranged in a specific order, but some of the objects are identical. For example, arranging a word with repeated letters or arranging a deck of cards with multiple identical cards.

5. Can the formula for arrangements with identical elements be used for non-integer values?

No, the formula for arrangements with identical elements can only be used for integer values. If the elements are not distinct, then the number of arrangements cannot be calculated using this formula.

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