Sure, I'd be happy to help you with these questions!
a) To calculate the total number of possible 6-digit security numbers, we can use the permutation formula:
nPr = n! / (n-r)!
where n is the total number of digits available (9) and r is the number of digits we are selecting (6).
So, the number of possible 6-digit security numbers without any restrictions is:
9P6 = 9! / (9-6)! = 9! / 3! = 9 x 8 x 7 x 6 x 5 x 4 = 60,480
Therefore, there are 60,480 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits.
b) If the first digit cannot be a one, we are left with 8 choices for the first digit (since we cannot use 1) and 8 choices for the second digit (since we have already used one digit). For the remaining 4 digits, we still have 7 choices for each digit, since we cannot repeat any digits.
Using the permutation formula again, the number of possible 6-digit security numbers with the first digit not being one is:
8 x 8 x 7 x 7 x 7 x 7 = 1,322,496
Therefore, there are 1,322,496 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the first digit is not one.
c) To create an odd number, the last digit must be an odd number, which means we have 5 choices for the last digit (1, 3, 5, 7, or 9). For the first digit, we cannot use 0 or 1, so we have 7 choices. For the remaining 4 digits, we still have 8 choices for each digit (since we can use any digit).
Using the permutation formula again, the number of possible 6-digit security numbers with the last digit being odd is:
7 x 8 x 8 x 8 x 8 x 5 = 7,1680
Therefore, there are 7,1680 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the last digit is odd.
d) To create a number greater than 300,000, the first digit must be 3, 4, 5, 6, 7, 8, or 9. If the first digit is 3, we have 7 choices for the first digit (3, 4, 5, 6, 7, 8, or 9). For the remaining 5 digits, we still have 8 choices for each digit.
If the first digit is not 3, we have 6 choices for the first digit (since we cannot use 1 or 2). For the remaining 5 digits, we still have 8 choices for each digit.
Using the permutation formula again, the number of possible 6-digit security numbers greater than 300,000 is:
7 x 8 x 8 x 8 x 8 x 8 + 6 x 8 x 8 x 8 x 8 x 8 = 2,526,720
Therefore, there are 2,526,720 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the number is greater than 300,000.
e) To create a number greater than 750,000, the first digit must be 8 or 9. If the first digit is 8, we have 2 choices for the first digit (8 or 9). For the remaining 5 digits, we still have 8 choices for each digit.
If the first digit is 9, we only have one choice for the first digit (9). For the remaining 5 digits, we still have 8 choices for each digit.
Using the permutation formula again, the number of possible 6-digit security numbers greater than 750,000 is:
2 x 8 x 8 x 8 x 8 x 8 + 1 x 8 x 8 x 8 x 8 x 8 = 262,144
Therefore, there are 262,144 possible 6-digit security numbers that can be made with the digits 1-9 without repeating any digits, where the number is greater than 750,000.