Combination and permutation are important mathematical
concepts used in calculating the probability of events. I have taken examples
from my statistical notes 8 and 9 to visualize the concepts of combination and
permutation with all objects selected without replacement and with replacement.
Combination is an arrangement of
objects in which the position of the objects does not matter. I will discuss about
it using two cases separately.
Combination of All Objects Selected
Without Replacement: In an example of statistical note 8 discussing on the
outcomes without replacement, there is only one combination of four balls, Red
(A), Blue (B), Green (C) and Yellow (D). Because every of 24 outcomes has all
four balls arranged in one way or the other. This is indicated by 1 number on
the right most part of the diagram 1 to mean one combination. This is a case of
Combination Without Replacement of All Objects. How many will be the outcomes
in case of Combination Without Replacement of some Objects from All Objects (Population)?
I will discuss this in other notes.
Diagram 1: Combination of All Objects Selected Without
Replacement
Combination of All Objects Selected With
Replacement:
In an example of statistical note 9 discussing on the outcomes with replacement,
there are 10 combinations of three balls, Blue (A), Green (B) and Red (C).
Because there is only one combination each of all three A balls, all three B balls
and all three C balls indicated by 1, 6 and 10 numbers respectively on the
right side of diagram 2. There are three combinations of two A balls and one B
ball, indicated by number 2, three combinations of two A balls and one C ball,
indicated by number 3, three combinations of one A ball and two B balls
indicated by number 4, six combinations of A BC ball in any order indicated by
number 5, three combinations of two B balls and one C ball indicated by number
7, three combinations of one B ball and two C balls indicated by number 8, three
combinations of one A ball and two C balls indicated by number 9.
Diagram 2: Combination of All Objects Selected With Replacement
Permutation is an arrangement of
objects in which the position of the objects matters. Ii is discussed using two
cases separately.
Permutation of All Objects Selected
Without Replacement: It was discussed in greater length in my statistical
note 8. Please refer to that note for detailed discussion. Here tree diagram 3 is
presented. In short, there are altogether 4! or 24 outcomes, each constituting four
balls selected without replacement. Every outcome is unique in terms of
position of four balls. This uniqueness of an outcome is because of the
position of each of four balls.
Diagram 3: Permutation of All Objects Selected Without
Replacement
Permutation of All Objects Selected With Replacement: It was discussed in greater length in my statistical note 9. Please refer
to that note for detailed discussion. Here tree diagram 4 is presented. In short,
there are altogether 33 or 27 outcomes, each constituting three balls selected with
replacement. Every outcome is unique in terms of position of three balls. This
uniqueness of an outcome is because of the position of each of three balls.
Diagram 4: Permutation of All Objects Selected With Replacement
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