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.
Important Questions from Permutation and Combination (Notes)
- In how many ways can 10 men be divided into two groups of 4 men and 6 men?
- Out of 5 consonants and 4 vowels, how many words of 3 consonants and 3 vowels can be made?
- How many 5-digit numbers can be formed from the digits 0, 2, 3, 4, 6, 7 and 9, using each at most once, which are divisible by 5?
- In how many distinguishable ways can the letters of the word CHANCE be arranged?
- From a group of 40 players, a cricket team of 11 players is chosen. Then, one of the eleven is chosen as the captain of the team. The total number of ways this can be done is
[$\binom{m}{n}$ below means the number of ways $n$ objects can be chosen from $m$ objects]
