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Digits Arrangement


AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
This item is taken from IGCSE Additional Mathematics (0606) Paper 22 of October/November 2010.
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To deal with this item, let us summarize the details of the problem:
   (a) Syllabus area: Permutation and Combination
 

   (b) Formula/Concept needed: 
            > Permutation or Combination
            > Fundamental Principle of Counting
            > Even numbers
            > No Repetition of Digits

Before we answer the items, let us encircle the important clues/phrases.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
It means that
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
The next phrase is
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
It means that
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Now, let us answer (i)
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Note that the number has four (4) digits
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Since there are no restrictions (except for the no repetition), we are considering all possible numbers that could be formed using the given digits. The first digit then can have
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
After using one on the first digit, the second digit will have
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
The third digit will then have
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
And the last digit will have
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Multiplying the options for each digit,
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Hence, there are 840 (4-digit) numbers that could be formed from the given options.

Another way of solving this item is to use Permutation (nPr). This is possible since order is important. That is
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers

For (ii)
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
To make sure that the number is less than 4000, we will start with the first digit. A number is less than 4000 if the first digit is a 1, 2, or 3. However, in the given options, there are only two (1 or 3).
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
The remaining could be any digit,
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Multiplying the options,
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Hence, there are 240 (4-digit) numbers that could be formed that are less than 4000.

For (iii)
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Again, to make sure that it is less than 4000, we will start with the first digit.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Further, to make sure that it is even, the last digit must contain an even digit. In the options given, only 4 and 8 are even.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers

The second and third may now have any of the remaining digits.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Multiplying the options,
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,digits,even numbers,numbers
Hence, there are 80 (4-digit) numbers that could be formed that are even and less than 4000.
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Thank you and God bless!  

2 comments:

  1. great explanation.wishing you can do that for all permtn and combntn questions.you are a great teacher

    ReplyDelete
    Replies
    1. Thank you for appreciating! I will be posting math videos explaining such items and more in my Youtube Channel: CIE Math Solutions. Don't forget to like, subscribe and share to others. Thank you!

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