※ Our check while writing this explanation could not fully confirm the published answer; scoring follows IPA's official key.
○ア
正解The correct answer is ア, 6,500 records. The calculation has three steps. First, find the maximum number of records that can be registered in Brand M from the customer code. The customer code is a 6-digit string made of the digits 0 to 9, but the 6th digit is a check digit, that is, an inspection digit determined by calculation from the values of the first 5 digits. Therefore only the first 5 digits can be assigned freely, so the number of records that can be registered is 10 to the 5th power = 100,000. Second, 5% of that is the number currently registered, so Brand M has 100,000 × 5% = 5,000 records. Third, find how many records are migrated from Brand S. Brand S has 2,000 records registered, but 10% of Brand M’s records, that is 5,000 × 10% = 500 records, are also registered in Brand S’s table as duplicates, and these are not migrated. So the number migrated from Brand S is 2,000 − 500 = 1,500. The number after consolidation is 5,000 + 1,500 = 6,500 records, so ア is correct.
×Incorrectイ
7,000 is the value you get by simply adding the two totals together without subtracting the duplicates: 5,000 + 2,000 = 7,000. However, condition (3) clearly states that 10% of Brand M’s records, that is 500 records, are already registered in Brand S’s table and are not migrated. To avoid counting the same customer twice, those 500 duplicate records must be subtracted.
×Incorrectウ
51,500 is the value you get if you mistakenly think the check digit position can also be assigned freely. Treating all 6 digits as free digits gives 10 to the 6th power = 1,000,000 records, 5% of which is 50,000, and adding the 1,500 records migrated from Brand S after subtracting the 500 duplicates gives 50,000 + 1,500 = 51,500. But the check digit is a dependent digit determined by the first 5 digits, so it does not increase the number of possible combinations.
×Incorrectエ
52,000 is the value you get by combining two mistakes: counting the check digit as a freely assignable digit, and not subtracting the duplicates. It takes 5% of 10 to the 6th power = 1,000,000 records to get 50,000, then adds Brand S’s 2,000 records as they are, giving 50,000 + 2,000 = 52,000. Because it contains two mistakes, it is the value farthest from the correct answer.
ここがポイント
A check digit is an inspection digit calculated from the other digits and appended in order to detect input errors; it is not a digit that can independently take any value. Therefore, if the 6th of 6 digits is a check digit, the number of items the code can identify is not 10 to the 6th power but 10 to the 5th power = 100,000 combinations. When estimating capacity from a coding scheme, the key point is not to multiply the check digit position into the number of combinations.
The number of records after a system consolidation (data migration) is found as: existing records at the destination + records at the source − records that exist in both (duplicates). This is the same idea as counting the elements of a union of sets (the inclusion-exclusion principle), and not double-counting duplicate data is exactly the criterion for judging whether the migration is correct. It is also important to read the subject of each condition carefully, so you do not mix up whether the 10% of duplicates is based on the destination (Brand M) or the source (Brand S).