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Using the digits {2, 3, . . . , 8}, find the number of different 5-digit numbers such that:
(a) Digits can be used more than once.
(b) Digits cannot be repeated, but can come in any order.
(c) Digits cannot be repeated and must be written in increasing order.
(d) Which of the above counting questions is a combination and which is a permutation? Explain
why this makes sense.

Respuesta :

As repetition is not allowed,

For first digit selection, we have 8 choices.

for second digit selection 7 choices.

Similarly, 6 choices for 3rd digit, 5 choices for 4th digit and 4 choices for 5th digit.

Hence, total choices 8 x 7 x 6 x 5 x 4 = 6720

MrRoyal

The 5-digit-numbers illustrate permutation and combinations.

  • There are 16807 ways to select the 5-digit numbers, if a digit can be used more than once
  • There are 2520 ways to select the 5-digit numbers, if a digit cannot be repeated
  • There are 1260 ways to select the 5-digit numbers, if a digit cannot be repeated and the number must be increasing order
  • Question (c) represents permutation while question (d) represents combination because order is important

The given parameters are:

[tex]\mathbf{n = 7}[/tex] --- count of digits 2 to 8

[tex]\mathbf{r = 5}[/tex] --- the number of digit

(a) Repetition is allowed

This means that, each of the 5 digit can be any of the 7 numbers.

So, the number of ways is:

[tex]\mathbf{Ways =7^5}[/tex]

[tex]\mathbf{Ways =16807}[/tex]

Hence, there are 16807 ways to select the 5-digit numbers, if a digit can be used more than once

(b) Repetition is not allowed

This means that, the order of the number is not important.

So, we make use of permutation

The number of ways is:

[tex]\mathbf{Ways = ^nP_r}[/tex]

[tex]\mathbf{Ways = ^7P_5}[/tex]

[tex]\mathbf{Ways = 7 \times 6 \times 5 \times 4 \times 3}[/tex]

[tex]\mathbf{Ways = 2520}[/tex]

Hence, there are 2520 ways to select the 5-digit numbers, if a digit cannot be repeated

(c) Repetition is not allowed, and must be increasing order

This means that, the order of the number is important.

So, we make use of combination

The number of ways is:

[tex]\mathbf{Ways = ^nC_r}[/tex]

[tex]\mathbf{Ways = ^7C_5}[/tex]

[tex]\mathbf{Ways = \frac{7 \times 6 \times 5 \times 4 \times 3}{2 \times 1}}[/tex]

[tex]\mathbf{Ways = 1260}[/tex]

Hence, there are 1260 ways to select the 5-digit numbers, if a digit cannot be repeated and the number must be increasing order

(d) Which represents permutation and combination?

  • Question (c) represents permutation because order is not important
  • Question (d) represents combination because order is important

Read more about permutation and combination at:

https://brainly.com/question/15301090

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