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Question

A software engineer creates a LAN game where an 8 digit code made up of 1,2,3,4,5,6,7,8 has to be decided on universal code. There is a condition that each number has to be used and no number can be repeated. What is the probability that first 4 digits of the code are even numbers ?


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Solution

Solve for the required probability:

There are 8 numbers to be filled in the 8 digit code without any repetition.

We know that probability PE=nEns where nE and ns are number of favorable outcomes and total number of possible outcomes respectively.

The number of possible arrangements of this are 8!.

Hence, the total number of outcomesns=8! ...(i)

Out of the given 8 numbers exactly 4 numbers are even2,4,6,8.

If the first 4 digits are to be occupied by even numbers then the last 4 digits must be occupied by odd numbers.

Now , the ways of arranging 4 even numbers in first 4digits are 4!

The ways of arranging 4 odd numbers in last 4digits are 4!

Hence, the favorable number of outcomesnE=4!×4! ...(ii)

PE=4!×4!8!

=248×7×6×5

PE=170

Hence, the probability that first 4 digits of the 8 digit code being even is 170.


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