Partial Permutation, Combination question for buying dogs at the pet store

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A person wants to buy 7 dogs from a pet store with specific requirements of at least 3 Chihuahuas and 2 Poodles. The initial calculation of combinations was incorrect due to double counting cases where more than the required number of Chihuahuas or Poodles were included. Clarification on whether the requirement is for at least or exactly 2 Poodles is needed for accurate calculations. The discussion highlights the complexity of calculating combinations without a spreadsheet, emphasizing the need to consider multiple cases separately. Ultimately, the problem requires careful consideration of distinct dog types and their combinations to avoid errors.
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TL;DR Summary: A pet store has 5 Chihuahua, 3 Fox and 4 Poodle. A person wants to buy 7 dogs. How many ways for the person to choose and buy with at least 3 Chihuahua and 2 Poodle.

There're C(5,3)*C(4,2) ways to buy dogs with at least 3 Chihuahua and 2 Poodle.
There're C(7,2) ways to buy dogs of any breeds.
In conclusion, there're C(5,3)*C(4,2)*C(7,2)=1260 ways for the person to pick and buy dogs.
Could someone check if my solution is correct? Thank you very much.
 
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Your solution is wrong.
 
PeroK said:
Your solution is wrong.
Could you be more specific? thanks.
 
Memo said:
Could you be more specific? thanks.
You are double/multiple counting cases where more than three Chihuahua's and/or more than 2 Poodles are bought.

Note: it's not clear to me whether the question is at least 2 Poodles or exactly 2 Poodles.
 
PeroK said:
You are double/multiple counting cases where more than three Chihuahua's and/or more than 2 Poodles are bought.

Note: it's not clear to me whether the question is at least 2 Poodles or exactly 2 Poodles.
I still don't get what you mean, could you point out whether it's the first or second part, or the whole part.
I think the question implies at least 2 Poodles
 
Memo said:
I still don't get what you mean, could you point out whether it's the first or second part, or the whole part.
I think the question implies at least 2 Poodles
How many ways are there to choose 7 dogs from 12 total?
 
PeroK said:
How many ways are there to choose 7 dogs from 12 total?
Ok, I see that my answer exceed the total possible ways. If 5 dogs are certain then they're 7 dogs left, and the person picks 2 out of 7 regardless of their breeds. How do I fix it?
 
Memo said:
Ok, I see that my answer exceed the total possible ways. If 5 dogs are certain then they're 7 dogs left, and the person picks 2 out of 7 regardless of their breeds. How do I fix it?
If we have at least 3 C's and at least 2 P's, then there are a number of different combinations that must be calculated separately. I don't see a quick way. I put the calculations on an Excel spreadsheet.

In any case, the possibilities are:
CPF
520
430
421
340
331
322

You can then, if you want, calculate all the other possibilities where there either ##C < 3## or ##P < 2## and check that the sum of all the possibilities is ##\binom{12} 7##.
 
PS that's why I thought the question might mean exactly two P's. Because that's simpler, as there are only three cases.
 
  • #10
I think you can group three Chihuahua's ##C_3##, and group 2 Poodles ##P_2##, leaving 2 dogs to select in any way.
 
  • #11
erobz said:
I think you can group three Chihuahua's ##C_3##, and group 2 Poodles ##P_2##, leaving 2 dogs to select in any way.
Not easily without multiple counting. That was the OP's initial error.
 
  • #12
PeroK said:
Not easily without multiple counting. That was the OP's initial error.
Yeah...never mind, not exactly straight forward.
 
  • #13
PeroK said:
If we have at least 3 C's and at least 2 P's, then there are a number of different combinations that must be calculated separately. I don't see a quick way. I put the calculations on an Excel spreadsheet.

In any case, the possibilities are:
CPF
520
430
421
340
331
322

You can then, if you want, calculate all the other possibilities where there either ##C < 3## or ##P < 2## and check that the sum of all the possibilities is ##\binom{12} 7##.
So would the answer be 6^3=216 ways? Like 5C 2P and 2P 5C would be considered the same but still in 2 options
 
  • #14
Memo said:
So would the answer be 6^3=216 ways? Like 5C 2P and 2P 5C would be considered the same but still in 2 options
You have to sum combinations for each specific case ( each row in the table ), I don't get 216 when I do that.
 
Last edited:
  • #15
Memo said:
So would the answer be 6^3=216 ways? Like 5C 2P and 2P 5C would be considered the same but still in 2 options
Not at all. There are six possibilities that are all different. You have to calculate them all.

Do you know how to use a spreadsheet?
 
  • #16
PeroK said:
Not at all. There are six possibilities that are all different. You have to calculate them all.

Do you know how to use a spreadsheet?
No, I can't use a spreadsheet while in exams. Is there really not a quick way since there were much harder questions waiting and if I solve it in the conventional ways then I just lose points
 
  • #17
May I suggest solving it as suggested (you don’t actually need a spreadsheet), then you can spend as much time as you like looking for a efficient method once you have the result.
 
  • #18
Memo said:
No, I can't use a spreadsheet while in exams
You won't have me in an exam.

Memo said:
. Is there really not a quick way since there were much harder questions waiting
I can imagine. The sooner you realise not all problems can be solved by a single calculation the better.
 
  • #19
I can see an example of multiple counting here( though can't think of how to deal with it).
So we have 12 dogs, 3 of type A: A1, A2, A3; similarly 4 of type B , 5 of type C. We want to select 7 dogs from the 12, so that we have 3 of type A, 2 of type B.
Consider this:

1)Choose A1,A2,A3 ;B1,B2; A4,A5

2) Choose A1,A4,A5; B1,B2; A2,A3

Both represent the choice , as a set:
{A1,A2,A3 A4,A5, B1,B2}

That's just one such duplicate.
 
  • #20
Are the Chihuahuas, Poodles, and Foxes distinct? That would make a difference. Then you have 5 slots spoken for - you only have 2 slots to fill with the remaining animals. If you have to distinguish which Poodles that you are using, it is more complex.
 

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