Binomial Expansion


AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals

This item is taken from Cambridge International AS and A Level Mathematics (9709) Pure Mathematics 1 Paper 11 of May/June 2010.

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To deal with this item, let us summarize the details of the problem:
   (a) Syllabus area: Series
  

   (b) Formula/Concept needed: 
            > Binomial Expansion
            > rth term or specific term of a binomial expansion
            > Product of two binomials

Recall the Binomial Expansion Theorem and the rth term of the Binomial Expansion.
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals

For part (i),
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
It means that
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
To do that, we need to identify the given values:
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
Substituting the values of a, b, n and r,
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
Simplify each of the terms
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals

Hence, the first three terms of the expansion is 
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals


For part (ii),
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
It means that
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
Take note that we are finding the
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
Since we are only interested in finding the coefficient of x, then
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
From part (i), we already have the term with x and x^3.
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals
Simplifying and combining like terms,
AS Level,A level,Exam,CIE,Cambridge,past papers,solutions,advance level,binomial expansion,product,powers,ascending power, descending power,first terms,binomial,series,sequences,Pascals

Hence, the coefficient of x in the product of the two binomials is 240. 

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Your comments and suggestions are welcome here. Write them in the comment box below.

Thank you and God bless!   

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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Your comments and suggestions are welcome here. Write them in the comment box below.


Thank you and God bless!  

Permutation or Combination


AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
This item is taken from Cambridge IGCSE Additional Mathematics (0606) Paper 11 of October/November 2015.
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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
            > Addition of Probabilities

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,choosing
It means that:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Next is
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
It means that:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
The first case is that there is only one (1) woman. It means that five (5) men are needed to complete the team of 6. The total number of possible selections are:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
The first case is that there are two (2) women. It means that four (4) men are needed to complete the team of 6. The total number of possible selections are:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
The first case is that there are three (3) women. It means that three (3) men are needed to complete the team of 6. The total number of possible selections are:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Adding the number of possible selections:
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Hence, there are 203 teams that could be formed with 6 members and at least one woman.

Another method is to consider all possible number of teams with 6 members that can be formed out of the 10. Here are the possible teams that could be formed without restrictions.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Let us separate the teams formed into two.
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Since we are only interested on the number of teams with at least one woman, we could deduct the number of teams without any woman from the total. That is
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
The total number of teams formed with 6 members (no restriction) is 
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
The total number of teams formed with 6 members without a woman is
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
or
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Subtracting them,
AS Level,Statistics,Exam,CIE,Mathematics,Probability,9709,revision,paper 6,past paper items,probability,Permutation,Combination,Selection,order,arrangement,choosing
Hence, the total number of teams formed with 6 members with at least one woman is 203.

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Your comments and suggestions are welcome here. Write them in the comment box below.


Thank you and God bless!  




IGCSE Mathematics (0580)

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AS/A Level Math (9709)

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