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ZIML Monthly Contest June-2022 Div M Question 10

 
 
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ZIML Monthly Contest June-2022 Div M Question 10
by Daniel He - Thursday, June 9, 2022, 11:55 AM
 

There are nine letters, "ENDOFYEAR", but there are two letters "E". Will the correct answer be:
5!*2= 240, instead of just 5! = 120?

Thanks.

 
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Re: ZIML Monthly Contest June-2022 Div M Question 10
by John Lensmire - Thursday, June 9, 2022, 2:37 PM
 

The two "E"s make things a little tricky here. Note that in a rearrangement the "E"s are identical, as swapping them doesn't change anything.

Let's look at a smaller example, were we can list everything. How many different rearrangements of "ROAR" are there that contain "OAR" in the sequence.

Here there's only $2! = 2$ (not $2!\cdot 2 = 4$) different rearrangements that contain "OAR":

  1. ROAR
  2. OARR

I think it also helps to list out ALL the different rearrangements of "ROAR" in total (note there are not $4! = 24$ rearrangements, there are $4!\div 2 = 12$ of them:

  1. AORR
  2. AROR
  3. ARRO
  4. OARR
  5. ORAR
  6. ORRA
  7. RAOR
  8. RARO
  9. ROAR
  10. RORA
  11. RRAO
  12. RROA

Hope this helps!