(a).

The company offers a home-security system that uses the numbers  through , inclusive, for a -digit security code.

Number of digits in the security code is .

There are digits for the code with  possibilities for each digit.

Therefore there are codes.

(b).

The order of the numbers in the code is important.

Condition : No digits can be repeated.

In this situation permutation of digits taken at a time.

The formula is .

Where is

is number of digits in the code i.e. .

Therefore security codes are available if no digits are repeated.

(c).

If the first digit is not or and the last digit should be .

Then there are  possibilities.

Since digits can be repeated, there are possibilities for the , , and digits.

There is only one possibility for the last digit, .

Thus, there are possible codes if the digits can be repeated.

If the first digit cannot be or , then there are  possibilities (Because the house owner wants, has to be the last digit).

Since digits cannot be repeated, there are just possibilities for the digit (Excluding first and last digit ).

There are then possibilities for the digit (The first two digits and the last digit, , are excluded)  possibilities for the , and possibility for the digit.

Therefore there are codes are available if no repetions are allowed.

(a).

codes.

(b).

codes.

(c).

codes.