SOLUTION: Determine the number of arrangements of the letters of the word SAMPLING a) if there are no restrictions b) if MP must not be together c) if the 2nd and 5th letters must b

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Question 1155284: Determine the number of arrangements of the letters of the word SAMPLING
a) if there are no restrictions
b) if MP must not be together

c) if the 2nd and 5th letters must be vowels
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.

            As this problem asks in its part  (a)  about the number of all possible arrangements,

            I make my conclusion,  that the visitor sees such a problem for the first time in his  (or her)  life.

            Therefore,  I will answer  ONLY  THIS  PART  (a),  since the other parts require more high level of knowledge/understanding. 


(a)  The word SAMPLING has 8 letters. All of them are unique, with no repeating.


     So, every arrangement is one of permutations of 8 letters, and every permutation is some unique arrangement.


    In other words, there is one to one correspondence between all distinguishable arrangements 

    and all possible permutations of 8 letters.


    The number of all possible permutations of 8 letters is 8! = 8*7*6*5*4*3*2*1 = 40320.    ANSWER

Part (a) completed and answered.

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

On Permutations,  see the lessons
    - Introduction to Permutations
    - PROOF of the formula on the number of Permutations
    - Problems on Permutations
    - Math circle level problem on Permutations

    - Arranging elements of sets containing indistinguishable elements
    - Persons sitting around a cicular table
    - OVERVIEW of lessons on Permutations and Combinations
in this site.

When you read these lesson and become familiar with the subject,  then the time will come for you to think on the parts  (b)  and  (c)  of your post.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Combinatorics: Combinations and permutations".


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.