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August 25th, 2013, 05:37 AM   #1
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How many ways a pack of cards can be cut into 3?

You are given a pack of 13 playing cards. A possible shuffling algorithm is to cut or
divide the pack into three parts, A, B, and C, and assemble the parts into a new order,
different from ABC. This process can then be repeated as many times as are needed to
obtain a sufficiently shuffled pack.
(a) In how many ways can the pack be cut into three, assuming that each part has at
least one card? Show your working.
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August 25th, 2013, 07:41 AM   #2
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Re: How many ways a pack of cards can be cut into 3?

Hello, maths4computing!

Quote:
You are given a pack of 13 playing cards.
A possible shuffling algorithm is to cut or divide the pack into three parts, A, B, and C,
and assemble the parts into a new order, different from ABC.
This process can then be repeated as many times as are needed to obtain a sufficiently shuffled pack.

(a) In how many ways can the pack be cut into three,
[color=beige]. . .[/color]assuming that each part has at least one card?

Deal the cards in a row, leaving a space between them.



There are 12 spaces.
Choose two of the spaces and insert "dividers".


Hence:








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August 25th, 2013, 07:58 AM   #3
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Re: How many ways a pack of cards can be cut into 3?

Thank you ever so much!

I knew there would be two selections where the cut off points were but wasn't too sure on how to go about it

Thanks again!
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August 25th, 2013, 08:03 AM   #4
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Re: How many ways a pack of cards can be cut into 3?

Quote:
Originally Posted by soroban
Hello, maths4computing!

Quote:
You are given a pack of 13 playing cards.
A possible shuffling algorithm is to cut or divide the pack into three parts, A, B, and C,
and assemble the parts into a new order, different from ABC.
This process can then be repeated as many times as are needed to obtain a sufficiently shuffled pack.

(a) In how many ways can the pack be cut into three,
[color=beige]. . .[/color]assuming that each part has at least one card?

Deal the cards in a row, leaving a space between them.



There are 12 spaces.
Choose two of the spaces and insert "dividers".


Hence:









So just out of interest how can they be put back together again other than the order ABC? I got 5 ways as 3! - 1 = 5 . Correct?
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August 25th, 2013, 02:47 PM   #5
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Re: How many ways a pack of cards can be cut into 3?

Hello again, maths4computing!

Quote:
So just out of interest, how can they be put back together again other than the order ABC?
I got 5 ways as: 3! - 1 = 5.[color=beige] .[/color]Correct?



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August 25th, 2013, 04:51 PM   #6
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Re: How many ways a pack of cards can be cut into 3?

66 includes stuff like 1-5-7 and 1-7-5.

If those are considered "same", then you have only 14 ways.
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