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The captain of a high school basketball team wants to assign unique three-digit number to all the team players. How many such unique numbers are possible if he decides to use only the even digits: 2, 4, 6 and 8 without repetition?
- a)10
- b)12
- c)18
- d)24
- e)32
Correct answer is option 'D'. Can you explain this answer?
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The captain of a high school basketball team wants to assign unique th...
Understand the objective
- Find the number of unique 3-digit numbers that can be formed using the numbers 2, 4, 6 and 8.
- (For example, numbers like 246, 468, 824 etc.)
- Note that repetition of these digits is not allowed while framing a unique 3-digit number.
- (For example, numbers like 226, 444, 884 etc. are not allowed)
- As we are arranging numbers here to form unique 3-digit numbers this is clearly a permutation problem.
- We can either use Filling Space method or Permutation Formula to solve this problem.
Method 1: Filling Space method
Write the objective equation enlisting all tasks
There is a total of four numbers: 2, 4, 6 and 8. To accomplish the objective of forming 3-digit numbers, three task needs to be done:
- Task1: Selecting a number for units’ digit
- This task can be achieved in 4 ways. (We can choose any of the four numbers)
- Task2: Selecting a number for tens digit.
- This task can be achieved in 3 ways.(We can choose among the remaining 3 numbers since we already selected one for the units’ digit)
- Task3: Selecting a number for hundreds digit.
- This task can be achieved in 2 ways. (We can choose among the remaining 2 numbers since we already selected two one for the units’ digit and other for the tens digit)
- All these three tasks must be done to form three-digit number.
Therefore,
The objective of forming unique 3-digit numbers can be found by multiplying the number of ways in which each task is achieved.
Therefore,
- No: of ways by which unique 3-digit numbers can be formed = 4 x 3 x 2 = 24ways.
Correct Answer: Option D
Method 2: Permutation Formula
We can also use the permutation formula (nPr) to find our answer.
Out of the given four numbers (2,4,6,8) we have to arrange three numbers so as to make a unique 3-digit number.
Therefore,
- n = 4 and r = 3
Hence, the number of ways by which unique 3-digit numbers can be formed
Correct Answer: Option D
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