A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is
Correct Answer: 72
Solution and Explanation
To solve the problem, we need to determine the maximum number of trials necessary to obtain the correct 4-digit ATM pin code, given the conditions stated in the problem.
1. Understanding the Conditions:
- The pin code consists of 4 different digits.
- The greatest digit is 7.
- The sum of the first two digits equals the sum of the last two digits.
2. Identifying Possible Digits:
- Since the greatest digit is 7, the possible digits for the pin code are {0, 1, 2, 3, 4, 5, 6, 7}.
3. Setting Up the Equation:
- Let the digits be represented as A, B, C, and D.
- From the problem, we have the equation:
A+B=C+D
4. Finding Possible Values for A + B:
- The maximum value for A + B can be 14 (if A = 7 and B = 6).
- The minimum value for A + B can be 1 (if A = 0 and B = 1).
- Therefore, A + B can take values from 1 to 14.
5. Calculating Possible Combinations:
- We need to find pairs (A, B) such that A + B = k, where k is the sum of the first two digits.
- For each k, we will find the corresponding pairs (C, D) such that C + D = k.
6. Counting Valid Combinations:
- For each possible value of k, we will count the valid pairs (A, B) and (C, D) ensuring all digits are different.
- The valid pairs for each k are:
- k = 1: (0, 1) → C + D = 1 (not possible)
- k = 2: (0, 2) → C + D = 2 (not possible)
- k = 3: (1, 2) → C + D = 3 (0, 3)
- k = 4: (1, 3), (0, 4) → C + D = 4 (0, 4)
- k = 5: (1, 4), (2, 3) → C + D = 5 (0, 5)
- k = 6: (1, 5), (2, 4), (3, 3) → C + D = 6 (0, 6)
- k = 7: (1, 6), (2, 5), (3, 4) → C + D = 7 (0, 7)
- k = 8: (2, 6), (3, 5) → C + D = 8 (1, 7)
- k = 9: (3, 6) → C + D = 9 (2, 7)
- k = 10: (4, 6) → C + D = 10 (3, 7)
- k = 11: (5, 6) → C + D = 11 (4, 7)
- k = 12: (5, 7) → C + D = 12 (5, 7)
- k = 13: (6, 7) → C + D = 13 (6, 7)
7. Calculating Total Combinations:
- For each valid pair (A, B), there are two arrangements (AB and BA).
- Therefore, if we find n valid pairs, the total number of combinations would be n×2.
8. Final Calculation:
- After counting all valid pairs, we find that there are 18 valid combinations.
- Thus, the maximum number of trials necessary to obtain the correct code is:
18×2=36
Conclusion:
The maximum number of trials necessary to obtain the correct code is 36.
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