Question

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 $2x_1x_2x_3x_4x_5x_6x_7$. An account number is considered to be a ‘magic’ number if $x_1x_2x_3$ is exactly the same as $x_4x_5x_6$ or $x_5x_6x_7$ or both. $x_i$ 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?

Accepted Answer

C. Case 1: $x_1x_2x_3 = x_4x_5x_6$. Total $= 10^3 	imes 10^1 = 10000$, minus invalid '0000000' gives 9999. Case 2: $x_1x_2x_3 = x_5x_6x_7$. Total $= 9999$. Intersection: $x_4=x_5=x_6=x_7$ and $x_1=x_2=x_3=x_5$. Total overlap $= 9$. Total $= 9999 + 9999 - 9 = 19989$.

Combinatorics: Magic Account Numbers

Explanation

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