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Outside a home, there is a 6-key keypad that can be used to open the garage if the correct four-digit code? is entered. (a) How many codes are possible? (b) What is the probability of entering the correct code on the first try, assuming that the owner doesn't remember the code?

Respuesta :

konrad509
a) [tex]6^4=1296[/tex]

b)
[tex]|\Omega|=1296\\ |A|=1\\\\ P(A)=\dfrac{1}{1296}\approx0.08\%[/tex]

Answer:

The total number of codes is found with

[tex]P_{n}^{r} =n^{r}[/tex]

This formula is applied when we have to find combinations where the order matters, and each element can be used more than once, as happens with codes, there can be repetitions of numbers.

So, [tex]n[/tex] represents the total number of elements which are 6, and [tex]r[/tex] represents the total digits per code, which are 4.

Replacing these values, we have

[tex]P_{6}^{4} =6^{4}=1296[/tex]

This means there are 1296 possible codes.

Now, if the owner doesn't remember the code, the probability of entering the correct code on the first try would be

[tex]P=\frac{1}{1296}=0.0008 (or \ 0.08\%)[/tex]

This means that it's nearly impossible to enter the right code at once.

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