Permutation and Combination-2

   In the previous post (Permutation and Combination-1), all four balls were different colors. However, in this post, we deal with the case that two balls are of the same color as shown in Fig. A-4. 

Fig. A-4 Two balls with the same color in four balls.

   In this case, how many arrangements can be distinguished by us if we line four balls up? By using the same method as Fig. A-3, we can obtain all possible ways to line four balls up as shown in Fig. A-5.

Fig. A-5 All possible 24 ways when two same colored balls are included.

   Although the indexing numbers of B1 and B2 are assigned in order to designate two blue balls, our eyes will perceive two blue balls as the same. In other words, we cannot distinguish each pair connected by arrows in Fig. A-5.
   Therefore, when four balls with two same color balls are lined up, we recognize that all possible ways to be distinguished are twelve as shown in Fig. A-6.

Fig. A-6 All possible 12 ways to be distinguished when two same colored balls are included.

   Furthermore, if three balls are the same color, we would say that all possible ways to be distinguished to us is only four as shown in Fig. A-7.

Fig. A-7 All possible 4 ways to be distinguished when three same colored balls are included.

   Finally, if all four balls are the same color, we know that there is only one distinguishable line up. Summarizing the discussion so far,

   (1) If four balls are all different colors and four balls are lined up in a row, all possible ways to be distinguished are 4!=24.

   (2) If two balls of four balls are the same color and four balls are lined up in a row, all possible ways to be distinguished are 24-12=12. We express this as twelve ways are redundancies or duplicated.

   (3) If three balls of four balls are the same color and four balls are lined up in a row, all possible ways to be distinguished are 24-20=4. We express this as twenty ways are redundancies or duplicated.

   (4) If four balls are the same color and four balls are lined up in a row, all possible way to be distinguished is only 1.

   If looking closely, we can see that the evaluated ways of (1), (2), (3) and (4) happen to be the same as 24=4!/0!, 12=4!/2!, 4=4!/3!, and 1=4!/4!, respectively.

   That is, the numerator's factorial corresponds to total number of balls and the denominator’s factorial corresponds to the number of balls with the same color. Therefore, when placing N balls with R same colored balls in a row, it can be easily inferred that all possible ways to be distinguished is simply N!/R!.

   Of course, there will be certainly any strict proof of N!/R!, but it may be sufficient that engineers simply use the formula of N!/R! leaving its proof to the mathematicians.

Permutation and Combination-1

   Now, there are four toy balls of red, yellow, blue and green, and assume the situation that a child is playing to arrange these four balls in a line. Let's think about how many ways he can line them up.

Fig. A-1 Four different colored balls; Red, Yellow, Blue and Green.

   In the playing with four balls, a child can arbitrary pick any of four colored balls up, and then place it on the first row. Therefore, a child have four ways for his first selection as shown in Fig. A-1.

Fig. A-2 Ways to place a ball in each trial.

   After the first choice, a child can also arbitrary pick one in the rested three balls up and place it on the second row, which means that a child has three choice to select a ball.
   If continuing this process to the last, a child will eventually be forced to chose the last remaining ball in the fourth trial.That is, 4 choices in the first trial; 3 choices in the second trial; 2 choices in the third trial; only 1 choice in the fourth trial.
   Then, it can be said that total possible ways to arrange four balls in a line are 24, which can be calculated from 4!=4x3x2x1 and called permutation.
   However, it should be noticed that all possible ways in the permutation are totally different arrangements, and therefore distinguishable as shown in Fig. A-3.

Fig. A-3 All possible 24 ways to line up four different colored balls.

Free PDF files of the textbooks used in universities

   So far, I have collected a lot of references from the internet through the electronic libraries of university or my web surfing for my studies and  researches.

   As you knows, since most students in least developed countries have suffered from the expensive price of the university textbooks in these days, I listed only the titles of textbooks used in universities, which can be personally provided by me.

   The listed textbooks are all PDF file format and opened in web sites. However, even if you use the listed ones for your personal study or learning, you can be faced with any copyright problems.

   If you massly distribute them around you, it would mean that you are seriously violating the copyright laws. Therefore, I listed only the titles of textbooks collected by me. Maybe, I am sure that you all knows the ways to use these files.

   You can freely download from various web sites just by copying and pasting the book title into Google searching window. If you can find your targeted PDF file, please contact with me.

(Engineering)

1. Fluid Mechanics: Fundamentals and Applications (6th Edition) by Yunus A. Cengel and John M. Cimbala, Published by McGraw-Hill Higher Education, 2006.
2. Fundamentals of Chemical Reaction Engineering by Mark E. Davis and Robert J. Davis, Published by McGraw-Hill, 2003.
3. Fundamentals of Electrical Engineering by Giorgio Rizzoni, Published by McGraw-Hill Higher Education, 2009.
4. Fundamentals of Heat and Mass Transfer (6th Edition) by F. P. Incopera, D. P. Dewitt, T. L. Bergman and A. Lavine, Published by John Wiley and Sons Inc., 2007.
5. Fundamentals of Thermodynamics (7th Edition) by C. Borgnakke and R. Sonntag, Published by John Wiley and Sons Inc., 2009.

(Mathematics)

1. Advanced Engineering Mathematics (10th Edition) by Erwin Kreyszig, Published by John Wiley and Sons Inc., 2011.
2. A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering (3rd Edition) by J. F. James, Published by Cambridge University Press, 2011.
3. Calculus by Gilbert Strang, Published by Wellesley-Cambridge Press, 2001.
4. Calculus (Volume 1) by Edwin Jed Herman and Gilbert Strang, Published by OpenStax, 2016.
5. Calculus (Volume 2) by Edwin Jed Herman and Gilbert Strang, Published by OpenStax, 2016.

   Since the above two calculus textbooks (4 and 5) by Edwin Jed Herman and Gilbert Strang are the literary works under CC BY-SA 4.0, you can freely download them without any concern on the copyright.

(Physics and Chemistry)

1. Atkins' Physical Chemistry (8th Edition) by Peter Atkins and Julio de Paula, Published by W. H. Freeman and Company, 2006.
2. Introduction to Solid State Physics (7th Edition) by Charles Kittel, Published by John Wiley and Sons Inc., 1996.
3. Introduction to Quantum Mechanics  by A. C. Phillipss, Published by John Wiley and Sons Inc., 2003.
4. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition) by Robert Eisberg, Published by John Wiley and Sons Inc., 1985.
5. Introduction to Spectroscopy: A Guide to  Students of Organic Chemistry (3rd Edition) by D. L. Pavia, G. M. Lampman and G. S. Kriz, Published by Brooks/Cole (Thomson Learning), 2001.