My Math Forum How many ways a pack of cards can be cut into 3?

 August 25th, 2013, 05:37 AM #1 Newbie   Joined: Jul 2013 Posts: 18 Thanks: 0 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.
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.

$\;\;\;\circ\_\circ\_\circ\_\circ\_\circ\_\circ\_\c irc\_\circ\_\circ\_\circ\_\circ\_\circ\_\circ$

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

Hence:

$\;\;\;\circ\,\circ\,|\,\circ\,\circ\,\circ\,\circ\ ,\circ\,\circ\,|\,\circ\,\circ\,\circ\,\circ\,\cir c\:\text{ represents }[2,\,6,\,5]$

$\;\;\;\circ\,\circ\,\circ\,\circ\,\circ\,\circ\,\c irc\,\circ\,|\,\circ\,|\,\circ\,\circ\,\circ\,\cir c \:\text{ represents }[8,\,1,\,4]$

$\text{Therefore, there are: }\:_{12}C_2 \:=\:\frac{12!}{2!\,10!} \:=\:66\text{ ways.}$

 August 25th, 2013, 07:58 AM #3 Newbie   Joined: Jul 2013 Posts: 18 Thanks: 0 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!
August 25th, 2013, 08:03 AM   #4
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Joined: Jul 2013

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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.

$\;\;\;\circ\_\circ\_\circ\_\circ\_\circ\_\circ\_\c irc\_\circ\_\circ\_\circ\_\circ\_\circ\_\circ$

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

Hence:

$\;\;\;\circ\,\circ\,|\,\circ\,\circ\,\circ\,\circ\ ,\circ\,\circ\,|\,\circ\,\circ\,\circ\,\circ\,\cir c\:\text{ represents }[2,\,6,\,5]$

$\;\;\;\circ\,\circ\,\circ\,\circ\,\circ\,\circ\,\c irc\,\circ\,|\,\circ\,|\,\circ\,\circ\,\circ\,\cir c \:\text{ represents }[8,\,1,\,4]$

$\text{Therefore, there are: }\:_{12}C_2 \:=\:\frac{12!}{2!\,10!} \:=\:66\text{ ways.}$

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?

August 25th, 2013, 02:47 PM   #5
Math Team

Joined: Dec 2006
From: Lexington, MA

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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?

$\;\;\;\;\text{Right!}$

 August 25th, 2013, 04:51 PM #6 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,103 Thanks: 907 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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