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Suppose we want to choose 5 colors, without replacement, from 18 distinct colors.

(a) If the order of the choices does not matter, how many ways can this be done?

(b) If the order of the choices matters, how many ways can this be done?

Suppose We Want To Choose 5 Colors Without Replacement From 18 Distinct Colors A If The Order Of The Choices Does Not Matter How Many Ways Can This Be Done B If class=

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CintiaSalazar

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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