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Proof, EXT2 P2 2017 HSC 16c

A 2 by grid is made up of two rows of square tiles, as shown.
 

The tiles of the 2 by grid are to be painted so that tiles sharing an edge are painted using different colours. There are different colours available, where  .

It is NOT necessary to use all the colours.

Consider the case of the 2 by 2 grid with tiles labelled A, B, C and D, as shown.
 

There are ways to choose colours for the first column containing tiles A and B. Do NOT prove this.

  1. Assume the colours for tiles A and B have been chosen. There are two cases to consider when choosing colours for the second column. Either tile C is the same colour as tile B, or tile C is a different colour from tile B.

     

    By considering these two cases, show that the number of ways of choosing colours for the second column is  .  (2 marks)

  2. Prove by mathematical induction that the number of ways in which the 2 by grid can be painted is  , for  .  (2 marks)

  3. In how many ways can a 2 by 5 grid be painted if 3 colours are available and each colour must now be used at least once?  (2 marks)

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Show Worked Solution
i.   

♦♦♦ Mean mark 20%.

 

 

 

ii.   

♦♦♦ Mean mark 22%.

 

 

 

 

iii.   

♦♦♦ Mean mark 22%.

 
 

 

Proof, EXT2 P2 2016 HSC 16c

In a group of people, each has one hat, giving a total of different hats. They place their hats on a table. Later, each person picks up a hat, not necessarily their own.

A situation in which none of the people picks up their own hat is called a derangement.

Let    be the number of possible derangements.

  1. Tom is one of the people. In some derangements Tom finds that he and one other person have each other's hat.

     

    Show that, for  , the number of such derangements is    (1 mark)

  2. By also considering the remaining possible derangements, show that, for  

      (2 marks)

  3. Hence, show that  ,  for    (1 mark)

  4. Given    and  ,  deduce that  ,  for    (1 mark)

  5. Prove by mathematical induction, or otherwise, that for all integers    (2 marks)

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Show Worked Solution

i.   

♦♦♦ Mean mark 22%.

 

ii.   

♦♦♦ Mean mark 6%.

 

 

 

♦♦ Mean mark part (iii) 37%.
iii.   
   

 

 

iv.  

♦♦♦ Mean mark 1%!

 

 

v.  

♦♦♦ Mean mark 13%.