Chapter 15.5, Problem 17WE

To determine

To calculate: The time taken to test all possible combinations to unlock a 3 number code.

Answer to Problem 17WE

The time taken to test all possible combinations to unlock a 3 number codeis $8568\text{ min}$ .

Explanation of Solution

Given information:

To unlock a 3 number code, digits available are from 0 to 35. It takes 12 seconds to test each combination. No repetition of digits is there.

Formula used:

Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Calculation:

Consider the provided information that to unlock a 3 number code, digits available are from 0 to 35. It takes 12 seconds to test each combination. No repetition of digits is there.

Let 3 number codebe $\boxed{}\boxed{}\boxed{}$ .

Since there are 36 digits available.

So, first number can be any 1 of 36 numbers available.

  $\boxed{36}\boxed{}\boxed{}$

As repletion of digits is not allowed.

So, second number can be chosen from remaining 35 numbers as 1 number is already used.

  $\boxed{36}\boxed{35}\boxed{}$

And, third number can be chosen from remaining 34 numbers as 2 numbersare already used.

  $\boxed{36}\boxed{35}\boxed{34}$

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $36\times 35\times 34=42840$

As each combination takes 12 seconds to test, so 42840 will take

  $\begin{array}{c}42840\times 12=514080\text{ seconds}\\ =\frac{514080}{60}\text{ minutes}\\ =8568\text{ minutes}\end{array}$

Thus, the time taken to test all possible combinations to unlock a 3 number code is $8568\text{ min}$ .