# How many possible arrangements are there for a deck of 52 playing cards?

• Benzoate

## Homework Statement

How many possible arrangements are there for a deck of 52 playing cards?(For simplicity, consider only the order of the cards , not whether they are turned upside down, etc.) suppose you start with a sorted deck and shuffle it repeatedly , so that all possible arrangements becomes accessible? how much entropy have you created in the process? express your answer both as a pure number(neglecting the factor k) and the SI units.Is the entropy significant compared to the entropy asociated with arranging thermal energy among the molecules in the cards?

## Homework Equations

S=k*ln(omega), k is neglected in this problem.

omega=(q+N)!/((q)!(N)!)

## The Attempt at a Solution

a)How many possible arrangements are there for a deck of 52 playing cards?

the number of arrangements is just N factorial or in my case 52!

b)suppose you start with a sorted deck and shuffle it repeatedly , so that all possible arrangements becomes accessible how much entropy have you created in the process?

so would I just calculate the total number of omega 's: In other words, would I calculate all the posible q's? for instance , omega(q=0)=(0+52)/((0!)(52!)+omega(q=1)=(1+52)/((1!)(52!))+...+omega(q=51)=(51+52)!/((51!)(52!))+omega(q=52)=(52+52)!/((52!)(52!)) and then proceed to take the natural log of all the total sums of the omega's to calculate my entropy?

I'm not sure where your equation for omega comes from, but omega is supposed to just be the number of possible arrangements. So your increase in entropy is just:

$$\Delta S = k \left( \ln 52! - \ln 1 \right)$$

genneth said:
I'm not sure where your equation for omega comes from, but omega is supposed to just be the number of possible arrangements. So your increase in entropy is just:

$$\Delta S = k \left( \ln 52! - \ln 1 \right)$$

Is my calculation for the total number of arrangements correct? I do not understand how you obtained ln 1.

i obtained my equation for omega on p. 63 of Daniel's V. Schroeder Thermal physics textbook.

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The number of arrangements is correct. The 1 comes from the fact that there is only one way to arrange the deck in a sorted manner.