Question 1147433
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<pre>

Word "PERMUTATIONS" has 12 symbols, of them "T" has multiplicity 2, and the rest symbols have multiplicity 1.


Therefore, the number of distinguishable arrangements of the word "PERMUTATIONS" is


    {{{N[1]}}} = {{{12!/2!}}} = {{{(12*11*10*9*8*7*6*5*4*3*2*1)/(2*1)}}} = 239500800.     (1)


In the formula (1), 2! in the denominator serves to account for repeating "T" in the word.




Word "COMBINATIONS" has 12 symbols, of them 

    - the letter "O" has multiplicity 2,

    - the letter "I" has multiplicity 2,

    - and all other letters are not repeating.


Therefore, the number of distinguishable arrangements of the word "COMBINATIONS" is


    {{{N[2]}}} = {{{12!/(2!*2!)}}} = {{{(12*11*10*9*8*7*6*5*4*3*2*1)/(1*2*1*2)}}} = 119750400.     (2)


In the formula (2), two factors  of  2!  in the denominator serves to account for repeating letters "O"  and  "I"  in the word.



The difference  {{{N[1]}}} - {{{N[2]}}} = 239500800 - 119750400 = 119750400.    <U>ANSWER</U>
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Solved, completed and explained in all details.


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On Permutations, &nbsp;see the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Introduction-to-Permutations.lesson>Introduction to Permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/PROOF-of-the-formula-on-the-number-of-permutations.lesson>PROOF of the formula on the number of Permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Problems-on-Permutations.lesson>Problems on Permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Arranging-elements-of-sets-containing-undistinguishable-elements.lesson>Arranging elements of sets containing indistinguishable elements</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Persons-sitting-around-a-circular-table.lesson>Persons sitting around a cicular table</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Combinatoric-problems-for-entities-other-than-permutations-and-combinations.lesson>Combinatoric problems for entities other than permutations and combinations</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Miscellaneous-problems-on-permutations-combinations-and-other-combinatoric-entities.lesson>Miscellaneous problems on permutations, combinations and other combinatoric entities</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-the-lessons-on-Permutations-and-Combinations.lesson>OVERVIEW of lessons on Permutations and Combinations</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Combinatorics: Combinations and permutations</U>". 



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.