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Q&A To correct for overcounting, why can't we divide by the sum of the over-counts? [duplicate]

0 answers  ·  posted 3y ago by DNB‭  ·  closed as duplicate 3y ago by Peter Taylor‭

Question combinatorics
#4: Question closed by user avatar Peter Taylor‭ · 2022-01-24T08:52:47Z (almost 3 years ago)
#3: Post edited by user avatar DNB‭ · 2022-01-24T08:05:19Z (almost 3 years ago)
  • Please see the sentence underlined in red below. Why not "2 + 2"?
  • >Problem 3.4: How many distinct arrangements are there of PAPA?
  • >
  • >![Image alt text](https://math.codidact.com/uploads/xVDVJFDdmSEaiw8LnsbcFxmv)
  • David Patrick, [BS Math & Computer Science, MS Math (Carnegie Mellon), PhD Math (MIT)](https://artofproblemsolving.com/wiki/index.php/David_Patrick). *Introduction to Counting & Probability* (2005), p 50.
  • The book doesn't expatiate the sentence underlined in red below. 1. Why "it isn't 2 + 2"?
  • 2. Note that $2 \times 2 = 2 + 2 = 4$! Is this a coincidence?
  • >Problem 3.4: How many distinct arrangements are there of PAPA?
  • >
  • >![Image alt text](https://math.codidact.com/uploads/xVDVJFDdmSEaiw8LnsbcFxmv)
  • David Patrick, [BS Math & Computer Science, MS Math (Carnegie Mellon), PhD Math (MIT)](https://artofproblemsolving.com/wiki/index.php/David_Patrick). *Introduction to Counting & Probability* (2005), p 50.
#2: Post edited by user avatar DNB‭ · 2022-01-24T08:02:54Z (almost 3 years ago)
  • Please see the sentence underlined in red below. Why not calculate overcounting as "2 + 2" ways?
  • >Problem 3.4: How many distinct arrangements are there of PAPA?
  • >
  • >![Image alt text](https://math.codidact.com/uploads/xVDVJFDdmSEaiw8LnsbcFxmv)
  • David Patrick, [BS Math & Computer Science, MS Math (Carnegie Mellon), PhD Math (MIT)](https://artofproblemsolving.com/wiki/index.php/David_Patrick). *Introduction to Counting & Probability* (2005), p 50.
  • Please see the sentence underlined in red below. Why not "2 + 2"?
  • >Problem 3.4: How many distinct arrangements are there of PAPA?
  • >
  • >![Image alt text](https://math.codidact.com/uploads/xVDVJFDdmSEaiw8LnsbcFxmv)
  • David Patrick, [BS Math & Computer Science, MS Math (Carnegie Mellon), PhD Math (MIT)](https://artofproblemsolving.com/wiki/index.php/David_Patrick). *Introduction to Counting & Probability* (2005), p 50.
#1: Initial revision by user avatar DNB‭ · 2022-01-24T08:02:35Z (almost 3 years ago) 
To correct for overcounting, why can't we divide by the sum of the over-counts?
Please see the sentence underlined in red below. Why not calculate overcounting as "2 + 2" ways? 

>Problem 3.4: How many distinct arrangements are there of PAPA? 
>
>![Image alt text](https://math.codidact.com/uploads/xVDVJFDdmSEaiw8LnsbcFxmv)

David Patrick, [BS Math & Computer Science, MS Math (Carnegie Mellon),  PhD Math (MIT)](https://artofproblemsolving.com/wiki/index.php/David_Patrick). *Introduction to Counting & Probability* (2005), p 50.
combinatorics