Considering the definition of permutation and combination:
- the number of ways of choosing 5 colors from 18 distinct colors (the order doesn't matter) is 8,568.
- the number of ways of choosing 5 colors from 18 distinct colors (the order of choices matters) is 1,028,160.
Definition of Permutation
Permutation is placing elements in different positions. So, permutations of m elements in n positions are called the different ways in which the m elements can be arranged occupying only the n positions.
In other words, permutations (or Permutations without repetition) are ways of grouping elements of a set in which:
- take all the elements of a set.
- the elements of the set are not repeated.
- order matters.
To obtain the total of ways in which m elements can be placed in n positions, the following expression is used:
[tex]mPn=\frac{m!}{(m-n)!}[/tex]
where "!" indicates the factorial of a positive integer, which is defined as the product of all natural numbers before or equal to it.
Definition of combination
Combinations of m elements taken from n to n (m≥n) are called all the possible groupings that can be made with the m elements in such a way that not all the elements enter; the order does not matter and the elements are not repeated.
To calculate the number of combinations, the following formula is applied:
[tex]C=\frac{m!}{n!(m-n)!}[/tex]
This case
In this case, you know we want to choose 5 colors, without replacement, from 18 distinct colors. This is:
- We want to choose 4 colors: n= 5
- The total number of distinct colors: m= 18
(a) If the order of the choices is not taken into consideration, you use a combination as follow:
[tex]C=\frac{18!}{5!(18-5)!}[/tex]
Solving:
[tex]C=\frac{18!}{5!13!}[/tex]
C= 8,568
Finally, the number of ways of choosing 5 colors from 18 distinct colors (the order doesn't matter) is 8,568.
(b) If the order of the choices matters, you use a permutation as follow:
[tex]18P5=\frac{18!}{(18-5)!}[/tex]
Solving:
[tex]18P5=\frac{18!}{13!}[/tex]
18P5= 1,028,160
Finally, the number of ways of choosing 5 colors from 18 distinct colors (the order of choices matters) is 1,028,160.
Learn more about
permutation:
brainly.com/question/12468032
brainly.com/question/4199259
combination:
brainly.com/question/25821700
brainly.com/question/25648218
brainly.com/question/24437717
brainly.com/question/11647449
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