Four-Digit Palindromic ATM PINs

An ATM PIN consists of digits. A four-digit PIN can be represented as $ABCD$, where A, B, C, and D are digits from 0 through 9.

The condition is that the PIN must read the same forwards and backward, meaning it must be a palindrome.

For a four-digit PIN $ABCD$ to be a palindrome, the following must be true:

• The first digit must match the last digit: $A = D$.
• The second digit must match the third digit: $B = C$.

Thus, any palindromic four-digit PIN must have the form $ABBA$.

Calculating Possible PIN Combinations

To find the total number of possible palindromic PINs, we need to count the independent choices for the digits.

• The first digit, $A$, can be any digit from 0 to 9. This gives 10 possible choices.
• The second digit, $B$, can also be any digit from 0 to 9. This gives 10 possible choices.
• The third digit, $C$, is determined by $B$ (since $C = B$).
• The fourth digit, $D$, is determined by $A$ (since $D = A$).

The total number of unique four-digit palindromic PINs is the product of the number of choices for $A$ and $B$.

Total Possible PINs = (Choices for $A$) $\times$ (Choices for $B$)

Total Possible PINs = $10 \times 10 = 100$.

Result

There are exactly 100 possible four-digit ATM PINs that are palindromic.