Combinatorics: Magic Account Numbers

XAT 2011 · Quantitative Ability & Data Interpretation · Medium · Permutations & Combinations

In a bank the account numbers are all 8 digit numbers, and they all start with the digit 2. So, an account number can be represented as 2x1x2x3x4x5x6x72x_1x_2x_3x_4x_5x_6x_7. An account number is considered to be a 'magic' number if x1x2x3x_1x_2x_3 is exactly the same as x4x5x6x_4x_5x_6 or x5x6x7x_5x_6x_7 or both. xix_i can take values from 0 to 9, but 2 followed by seven 0s is not a valid account number. What is the maximum possible number of customers having a 'magic' account number?

  1. A.

    9989

  2. B.

    19980

  3. C.

    19989

  4. D.

    19999

  5. E.

    19990

Answer

C

Explanation

Case 1: x1x2x3=x4x5x6x_1x_2x_3 = x_4x_5x_6. Total =103×101=10000= 10^3 \times 10^1 = 10000, minus invalid '0000000' gives 9999. Case 2: x1x2x3=x5x6x7x_1x_2x_3 = x_5x_6x_7. Total =9999= 9999. Intersection: x4=x5=x6=x7x_4=x_5=x_6=x_7 and x1=x2=x3=x5x_1=x_2=x_3=x_5. Total overlap =9= 9. Total =9999+9999−9=19989= 9999 + 9999 - 9 = 19989.

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