P&C -12 (XAT-2011) (3 Marks)

In a bank the account numbers are all 8 digit numbers, and they all start with digit 2. So an account number can be represented as 2x₁x₂x₃x₄x₅x₆x₇. An account number is considered to be a ‘magic’ number if x₁x₂x₃ is exactly same as x₄x₅x₆ or x₅x₆x₇ or both. x_i can take values from 0 to 9, but 2 followed by seven zeroes is not a valid account number. What is the maximum possible number of customers having a ‘magic’ account number?

Solution follows here:

Solution:

2x₁x₂x₃x₄x₅x₆x₇

Considering the case “x₁x₂x₃ is exactly same as x₄x₅x₆”:

x₁x₂x₃= x₄x₅x₆ = 000, x₇ = 1 to 9 as ‘20000000’ is not valid => 9 possibilities

x₁x₂x₃ = x₄x₅x₆ = 001 to 999, x₇ = 0 to 9 => 999*10 = 9990 possibilities

Considering the case “x₁x₂x₃ is exactly same as x₅x₆x₇”:

x₁x₂x₃= x₅x₆x₇ = 000, x₄ = 1 to 9 as ‘20000000’ is not valid => 9 possibilities

x₁x₂x₃ = x₅x₆x₇ = 001 to 999, x₄ = 0 to 9 => 999*10 = 9990 possibilities

Subtracting common possibilities in both of the above cases:

x₄x₅x₆ = x₅x₆x₇ => x₄ = x₅, x₅ = x₆, x₆ = x₇ => x₄ = x₅ = x₆ = x₇

=> excepting all zero case, the possibilities are 1111,2222,...,9999 => 9 possibilities

These 9 cases are repeated in both of the above cases, hence we need to subtract ‘9’ from the sum of no. Of possibilities of both the cases

=> Answer = 9 + 9990 + 9 + 9990 – 9 = 19989

Answer (3)