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# Combinations.....

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Arrange the five letters, ABCDE, into a group of 3-letter "words" with repeats of the letters allowed, such as AAA, AAB, AAC.....etc. How many such 3-letter combinations are possible? Thank you.

May 18, 2018

#1
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These are combinations with repeats allowed. So, we have:

[5 + 3 - 1]C3 =7C3 = 35. Here are all your combinations:

{A, A, A} | {A, A, B} | {A, A, C} | {A, A, D} | {A, A, E} | {A, B, B} | {A, B, C} | {A, B, D} | {A, B, E} | {A, C, C} | {A, C, D} | {A, C, E} | {A, D, D} | {A, D, E} | {A, E, E} | {B, B, B} | {B, B, C} | {B, B, D} | {B, B, E} | {B, C, C} | {B, C, D} | {B, C, E} | {B, D, D} | {B, D, E} | {B, E, E} | {C, C, C} | {C, C, D} | {C, C, E} | {C, D, D} | {C, D, E} | {C, E, E} | {D, D, D} | {D, D, E} | {D, E, E} | {E, E, E} (total: 35)

May 18, 2018
#2
+102763
+1

There are 5 choices for the first letter, 5 for the second and 5 for the third so that is  5^3 = 125 words

May 19, 2018
#3
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Melody: What you calculated are "permutations with repeats allowed" as compared to the first answer, which are "combinations with repeats allowed". Here are the "permutations" you calculated:

{A, A, A} | {A, A, B} | {A, A, C} | {A, A, D} | {A, A, E} | {A, B, A} | {A, B, B} | {A, B, C} | {A, B, D} | {A, B, E} | {A, C, A} | {A, C, B} | {A, C, C} | {A, C, D} | {A, C, E} | {A, D, A} | {A, D, B} | {A, D, C} | {A, D, D} | {A, D, E} | {A, E, A} | {A, E, B} | {A, E, C} | {A, E, D} | {A, E, E} | {B, A, A} | {B, A, B} | {B, A, C} | {B, A, D} | {B, A, E} | {B, B, A} | {B, B, B} | {B, B, C} | {B, B, D} | {B, B, E} | {B, C, A} | {B, C, B} | {B, C, C} | {B, C, D} | {B, C, E} | {B, D, A} | {B, D, B} | {B, D, C} | {B, D, D} | {B, D, E} | {B, E, A} | {B, E, B} | {B, E, C} | {B, E, D} | {B, E, E} | {C, A, A} | {C, A, B} | {C, A, C} | {C, A, D} | {C, A, E} | {C, B, A} | {C, B, B} | {C, B, C} | {C, B, D} | {C, B, E} | {C, C, A} | {C, C, B} | {C, C, C} | {C, C, D} | {C, C, E} | {C, D, A} | {C, D, B} | {C, D, C} | {C, D, D} | {C, D, E} | {C, E, A} | {C, E, B} | {C, E, C} | {C, E, D} | {C, E, E} | {D, A, A} | {D, A, B} | {D, A, C} | {D, A, D} | {D, A, E} | {D, B, A} | {D, B, B} | {D, B, C} | {D, B, D} | {D, B, E} | {D, C, A} | {D, C, B} | {D, C, C} | {D, C, D} | {D, C, E} | {D, D, A} | {D, D, B} | {D, D, C} | {D, D, D} | {D, D, E} | {D, E, A} | {D, E, B} | {D, E, C} | {D, E, D} | {D, E, E} | {E, A, A} | {E, A, B} | {E, A, C} | {E, A, D} | {E, A, E} | {E, B, A} | {E, B, B} | {E, B, C} | {E, B, D} | {E, B, E} | {E, C, A} | {E, C, B} | {E, C, C} | {E, C, D} | {E, C, E} | {E, D, A} | {E, D, B} | {E, D, C} | {E, D, D} | {E, D, E} | {E, E, A} | {E, E, B} | {E, E, C} | {E, E, D} | {E, E, E} (total: 125)

Note: See the difference between the two in this article:

https://en.wikipedia.org/wiki/Combination.  [Look under "Number of combinations with repetition"

May 19, 2018
edited by Guest  May 19, 2018
#4
+102763
+1

Yes I know what I answered.

I answered the question that was asked.

Melody  May 21, 2018