There are several ways the door can be locked, these ways illustrate combination.
There are 3375 possible combinations
From the question, we have:
[tex]\mathbf{n = 5}[/tex] --- the number of digits
[tex]\mathbf{r = 3}[/tex] ---- the number of actions
Each of the three actions can either be:
The number of ways of pressing a button is:
[tex]\mathbf{n_1 = ^5C_1}[/tex]
Apply combination formula
[tex]\mathbf{n_1 = \frac{5!}{(5-1)!1!}}[/tex]
[tex]\mathbf{n_1 = \frac{5!}{4!1!}}[/tex]
[tex]\mathbf{n_1 = \frac{5 \times 4!}{4! \times 1}}[/tex]
[tex]\mathbf{n_1 = 5}[/tex]
The number of ways of pressing a pair is:
[tex]\mathbf{n_2 = ^5C_2}[/tex]
Apply combination formula
[tex]\mathbf{n_2 = \frac{5!}{(5-2)!2!}}[/tex]
[tex]\mathbf{n_2 = \frac{5!}{3!2!}}[/tex]
[tex]\mathbf{n_2 = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1}}[/tex]
[tex]\mathbf{n_2 = 10}[/tex]
So, the number of ways of performing one action is:
[tex]\mathbf{n =n_1 + n_2}[/tex]
[tex]\mathbf{n =5 + 10}[/tex]
[tex]\mathbf{n =15}[/tex]
For the three actions, the number of ways is:
[tex]\mathbf{Action = n^3}[/tex]
[tex]\mathbf{Action = 15^3}[/tex]
[tex]\mathbf{Action = 3375}[/tex]
Hence, there are 3375 possible combinations
Read more about permutation and combination at:
https://brainly.com/question/4546043
