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: 0, 2, 4 and 6 without repetition?a)10b)12c)18d)24e)32Correct answer is option 'C'. Can you explain this answer?

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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: 0, 2, 4 and 6 without repetition?

a)
10
b)
12
c)
18
d)
24
e)
32

Correct answer is option 'C'. Can you explain this answer?

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Test: Permutation And Combination- 1

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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 0, 2, 4 and 6.
- (For example, numbers like 206, 460, 624 etc.)

Remember, we cannot use ‘0’ in the hundreds digit as we are asked to form three-digit numbers.
- (For example, 024, 046 are not three-digit numbers and hence not valid choices in this problem)

Also, note that repetition of these digits is not allowed while framing a unique 3-digit number.
- (For example, numbers like 226, 444, 200, etc. are not allowed)

As we are arranging numbers here to form unique 3-digit numbers this is clearly a permutation problem.

We can 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: 0, 2, 4 and 6

Similar to the previous problem, to accomplish the objective of forming 3-digit numbers, three task needs to be done:

Since we have a restriction of not using ‘0’ in the hundreds digit, we should first deal with that restriction.

Therefore,

· Task1: Selecting a number for hundreds digit

o This task can be achieved in 3 ways.

(We can choose from any 3 numbers: 2, 4 or 6. As we need a 3-digit number we cannot use ‘0’ at this place)

· Task2: Selecting a number for tens digit

o This task can be achieved in 3 ways.

(We can choose among the remaining 3 numbers since we already selected one for the hundreds digit)

· Task3: Selecting a number for units’ digit

o This task can be achieved in 2 ways.

(We can choose among the remaining 2 numbers since we already selected two: one for the hundreds 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 = 3 x 3 x 2 = 18 ways.

Correct Answer: Option C

Method 2: Permutation Formula

We can also use the permutation formula (nP_r) to find our answer.

Out of the given four numbers (0, 2,4,6) we have to arrange three numbers so as to make a unique 3-digit number.

We also know that we cannot use 0 as hundreds digit.

For the time being, let us consider that ‘0’ can be the hundreds digit.

If we also include ‘0’ as hundreds digit, then we are simply arranging three numbers from a list of four numbers.

The number of possibilities in that case would be 4P₃= 24

Now we know that among these 24 unique numbers, some numbers are those with ‘0’ as the hundreds digit. So, we need to subtract those.

If ‘0’ is fixed as the hundreds digit, then for the tens and units digit (2 places) we have the option to choose from remaining 3 numbers (2,4,6).

This can be done in 3P₂ ways = 6ways

That is, out of the 24 possibilities, 6 are those numbers with ‘0’ as the hundreds digit.

Therefore,

Number of unique three-digit numbers using digits 0,2,4 and 6 = 24 – 6 = 18

Correct Answer : Choice C

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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: 0, 2, 4 and 6 without repetition?a)10b)12c)18d)24e)32Correct answer is option 'C'. Can you explain this answer?

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