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?
3720087
To determine the number of different passwords Caleb can create, we need to consider the possible positions of the number 00 within an 8-digit password.
As the number 00 must be used exactly once, there are seven remaining positions to fill with digits other than 0. Since each position can be filled with any of the 10 digits from 0 to 9 (excluding 00), we can calculate the number of different passwords as follows:
Number of possible passwords = 7 * (10^7) = 70,000,000
Therefore, Caleb can create 70 million different passwords if he wants to use 00 exactly once.
To calculate the number of different passwords Caleb can create, we need to consider the total number of possible combinations.
Since Caleb wants to use "00" exactly once, we can break down the problem into two parts:
1. Determine the placement of "00" within the password.
2. Calculate the remaining digits that can be used to fill the remaining positions.
Step 1: Placement of "00"
To determine the placement of "00" within the password, we need to consider the available positions. Since "00" is a 2-digit group, we have 7 possible positions (1, 2, 3, 4, 5, 6, 7) where we can place it in an 8-digit password.
Therefore, there are 7 different ways to place "00" within the password.
Step 2: Remaining digits
Once we have determined the placement of "00," we need to calculate the remaining digits that can be used to fill the remaining positions. Since Caleb can use any number (0-9) excluding "00," there are 10 - 1 = 9 possibilities for each remaining position. Since we have 6 remaining positions after placing "00," the total number of possible combinations is 9^6.
Therefore, there are 9^6 = 531,441 different combinations for the remaining digits.
Step 3: Total number of passwords
To calculate the total number of passwords, we multiply the number of ways to place "00" by the number of possible combinations for the remaining digits:
Total number of passwords = Number of ways to place "00" * Number of combinations for remaining digits
= 7 * 9^6
= 3,720 * 9^6
= 3,720 * 531,441
= 1,975,313,520
Therefore, there are a total of 1,975,313,520 different passwords Caleb can create using only numbers, with "00" appearing exactly once.