Permutations

 

 The product rule for counting  

 

• The product rule for counting is used to find the number of possible terms for a given criteria 

• For example: 

  • How many 5-digit odd numbers can be formed using the digits 1  3  4  6  8  with no repetition of any digit 

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Example Question:

 

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• We start by drawing out 5 boxes as Jackson makes a 5 digit number 

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• Now we need to figure out what number goes in each box 

• As we told that the numbers that Jackson makes are greater than 50,000, we put the number 3 in the first box as there are only 3 numbers that can make a number greater than 50,000( 5, 8, 9) 

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• Now as there are no more criteria's for Jackson's number, we now must use the remain numbers up 

• So in the second box we put the number 4 as we have 4 cards left as we have already used 1 

• In the third box we put 3 as there, in the fourth box we put 2 and the last box we put 1 

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• Now we multiply all these numbers together in order to find the answer, which comes out as  

 

              3 x 4 x 3 x 2 x 1 = 72 

 

Practice Question: 

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Solution:

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Surds  

 

 

How can we simplify surds: 

 

• There are two methods that we can use to simplify surds 

• So let's consider the surd √300 

 

Method 1: 

 

• We can simply a surd by spotting its factors 

• For √300 we can see that it is equal to √100 x √3 

• And we know that √100 = 10  

• So √300 = 10√3 

 

Method 2: 

 

• We can also use prime factors in order to simplify a surd 

• For √300, we can find all the prime factors of 300: 

  • 300 = 2x2x3x5x5 

• Here we look for any repetitions, so for 300 we can see that the prime factors 2 and 5 are repeated 

• This means we can pull out 2x5, leaving the 3 behind to get 2x5√3, which equals 10√3 

• Therefore √300 = 10√3 

 

 

Practice Questions: 

 

1. √150  

2. √8 + √18 

3. Write √200 - √72 + 3√162 in the form of x√2 

 

 

Answers: 

 

   1. √150 = √2x3x5x5 = 5√6 ​​

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   2. √8 = √4 x √2 = 2√2 

       √18 = √9 x √2 = 3√2 

       2√2 + 3√2 = 5√2 

 

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   3. √200 = √100 x √2 = 10√2 

       √72 = -√9 x √8 = -3 x 2√2 = -6√2 

       3√162 = 3(√81 x √2) = 3(9√2) = 27√2 

 

       10√ - 6√2 + 27√2 = 31√2 where x = 31 

 

 

Rationalising Denominator: 

 

• Sometimes we get surds in the form of fractions 

• In order to simplify these we must rationalise the denominator 

• This means multiplying the fraction we have by another fraction that means the denominator becomes a rational number 

• For example if we had the fraction: 

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• In order to rationalise this we must multiply it by the fraction:

 

 

 

 

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• We use this fraction as it forms a rational denominator but will not change the original fraction as it is equal to 1 

• So we have: 

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• This will get us the result: 

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• And hence the denominator has been rationalised 

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Practice Questions: 

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Solutions:

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