Question 1179275
.
Calculate the total number of different permutations of all letters A, B, C, D, E, F when
a) there are no restrictions,
b) the letters A and B are to be adjacent to one another,
c) the first letter is A, B or C and the last letter is D, E or F.
~~~~~~~~~~~~~~~~


<pre>
(a)  6! = 6*5*4*3*2*1 = 720.



(b)  You concider the pair of letters AB as one glued object;

     so you have  5 object to permutate, in all, which produces 5! = 5*4*3*2*1 = 120 different permutations.


     Another 120 different permutations you have with the object BA instead of AB.


     So, in total, there are  120 + 120 = 240 such permutations.



(c)  You have 3 possibility for the first letter (A, B or C).

     Similarly, you have 3 possibility for the last letter (D, E or F).

     Finally, you have 4! = 4*3*2*1 = 24 permutations for the remaining 6-2 = 4 letters.


     Hence, in all, there are  3*3*24 = 216  possible permutations, satisfying the imposed conditions.
</pre>

Solved.


----------------


On Permutations, &nbsp;see introductory 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>Simple and simplest problems on permutations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Special-type-permutations-problems.lesson>Special type permutations problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/How-many-different-permutations-may-exist-ubder-given-restrictions.lesson>Problems on Permutations with restrictions</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.