Combination usage in the well-known word problem

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Discussion Overview

The discussion centers on the problem of determining the number of different letter arrangements that can be made from the letters of the word "MISSISSIPPI" using combinations rather than permutations. Participants explore the implications of using combinations in this context and the distinction between combinations and permutations.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the application of combinations in the context of the problem, specifically how it relates to avoiding repeated arrangements of letters.
  • Another participant explains that combinations refer to selecting items where order does not matter, contrasting this with permutations where order is significant.
  • A participant argues that the provided answer represents permutations rather than combinations, emphasizing that different arrangements of the same letters are not considered different combinations.
  • There is a suggestion to label the duplicated letters to illustrate the difference between permutations and combinations, indicating that labeled letters yield distinct permutations while unlabeled letters do not.

Areas of Agreement / Disagreement

Participants express differing views on the correct application of combinations versus permutations in solving the problem. There is no consensus on the appropriate method to use for counting the arrangements.

Contextual Notes

The discussion highlights the confusion surrounding the definitions and applications of combinations and permutations, particularly in the context of repeated elements in a set. The mathematical steps and reasoning behind the proposed combinations are not fully resolved.

Who May Find This Useful

This discussion may be of interest to individuals studying combinatorics, particularly those grappling with problems involving arrangements of letters or objects with repetitions.

M. next
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How many different letter arrangement can be made from the 11 letters of MISSISSIPPI?

(But using COMBINATION not the different permutation method)

I saw an answer and it says:
(combination of 1 out of 11)*(combination of 4 out of 10)*(combination of 4 out of 6) *(combination of 2 out of 2)

What does combination exactly mean, and in what way was it used here?

I mean if we don't want same words to appear, what will this have to do with taking say 4 Ss out of the left 10?

Thanks in advance.
 
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Hi M. next! :smile:
M. next said:
What does combination exactly mean, and in what way was it used here?

I mean if we don't want same words to appear, what will this have to do with taking say 4 Ss out of the left 10?

"combination of k out of n", also written "n choose k", or nCk, or the vertical matrix (n k), and equal to n!/k!(n-k)!,

is the number of ways of selecting k things out of n where the order does not matter

 
M. next said:
(But using COMBINATION not the different permutation method)

I saw an answer and it says:
(combination of 1 out of 11)*(combination of 4 out of 10)*(combination of 4 out of 6) *(combination of 2 out of 2)Thanks in advance.
That's the answer for the number of permutations, not the number of combinations.

The difference between the two: "issi" and "siis" are two different permutations (arrangements) of the letters i,i,s, and s. They are not different combinations, however; order doesn't matter in combinations.

Suppose we put labels on the duplicated letters, making mississippi become mi1s1s2i2p1p2i3s3s4i4. There are 11 factorial (11!) permutations of these labeled letters because each (labeled) letter is distinct. For example, i1s1s2i2 and i1s1s3i2 are distinct permutations. Take those labels away and these two permutations become issi and issi: they are now indistinguishable.

The problem is to find the number of distinguishable permutations of the given (unlabeled) letters.
 
oh thanks, and sorry for the late reply
 

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