XAT 2011 — QA & DI Question 31
Answer the following question based on the information given below.
From a group of 545 contenders, a party has to select a leader. Even after holding a series of meetings, the politicians and the general body failed to reach a consensus. It was then proposed that all 545 contenders be given a number from 1 to 545. Then they will be asked to stand on a podium in a circular arrangement, and counting would start from the contender numbered 1. The counting would be done in a clockwise fashion. The rule is that every alternate contender would be asked to step down as the counting continued, with the circle getting smaller and smaller, till only one person remains standing. Therefore the first person to be eliminated would be the contender numbered 2.
ln 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 2x1x2x3x4x5x6x7. An account number is considered to be a ‘magic’ number if x1x2x3 is exactly the same as x4x5x6, or x5x6x7 or both. xi 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?
Answer & solution
- A
9989
- B
19980
19989
- D
19999
- E
19990
x1x2x3x4x5x6x7 can be of the form abcdabc or abcabcd.
abc can be chosen in 10 × 10 × 10 = 1000 ways.
d can be chosen in 10 ways.
∴ x1x2x3x4x5x6x7 can be chosen in 2 × 1000 × 10 = 20000 ways.
However, this includes the ways in which all of a, b, c and d are 0, 1, 2, 3, …, 9, twice.
We subtract these numbers to get 20000 – 10 = 19990
We also don’t want 2000000.
∴ The total number of ways = 19989.
Hence, option (c).