I'm a bit stuck on this question:
Caleb needs to create an 8-digit password using only numbers. How many different passwords are there if he wants to use 00 exactly once?
There's no answer choices, I'm supposed to show all my work. Please help!
this problem is discussed many time.
Try google.
Start by placing the 0 0 side by side...
_ _ _ _ _ _ _ _ so the 0 0 could go
0 0
0 0
0 0
0 0
0 0
0 0
0 0
So the 0 0 could be in 7 places. So place them first (they could fill any of 7 spots... so they are 7)
7 _ _ _ _ _ _ which means there are 6 other spots to fill
now the other spots could each have the numbers 1-9, that is 10 numbers...
7 x 10 _ _ _ _ _ but then the condition says that the numbers have to be different ... so once you have used a digit it is no longer available. So in the next spot there are only 9 choices etc
7 x 10 x 9...
I hope this is enough of a hint to get you started : )
Oh drat! The program didn't show the 0 0 being shifted over to the next two spots in each of the lines beneath the first one.
0 0 _ _ _ _ _ _
_ 0 0 _ _ _ _ _
_ _ 0 0 _ _ _ _
_ _ _ 0 0 _ _ _
etc...
To solve this problem, we can break it down into several steps:
Step 1: Determine the position of '00' in the password.
Since Caleb wants to use '00' exactly once, we need to determine where it can be placed in an 8-digit password. '00' can be positioned in any of the 7 spaces between the digits or at the beginning or end of the password.
Step 2: Calculate the number of possibilities for the remaining digits.
After placing '00' in one of the positions, we need to fill the remaining 6 places with digits from 0 to 9 (excluding '00' as it is already used). Each of the remaining 6 places can be filled independently, so we calculate the possibilities for each place and multiply them together.
Since there are 10 possible digits (0-9), the number of possibilities for each remaining place is 10 - 1 (excluding '00').
Step 3: Multiply the number of possibilities together.
Since each step is independent of the others, we can multiply the number of possibilities for each step together to get the total number of different passwords.
Let's calculate it step by step:
Step 1: Determine the position of '00' in the password.
- '00' can be positioned in any of the 7 spaces between the digits or at the beginning or end of the password.
- So, there are a total of 7 + 2 = 9 possible positions for '00'.
Step 2: Calculate the number of possibilities for the remaining digits.
- After placing '00' in one of the positions, we need to fill the remaining 6 places with digits from 0 to 9 (excluding '00').
- Each of the remaining 6 places can be filled independently, so there are a total of (10 - 1) * (10 - 1) * ... * (10 - 1) = (10 - 1)^6 possibilities.
- This can also be written as 9^6.
Step 3: Multiply the number of possibilities together.
- Since each step is independent, we multiply the number of possibilities for each step together.
- The total number of different passwords is 9 * 9^6 = 9^7.
Therefore, there are a total of 9^7 = 478,296 different passwords that Caleb can create if he wants to use '00' exactly once and the password is 8 digits long.