It is known that    for all integers such that  . (Do NOT prove this.)

Find ONE possible set of values for and such that

  (2 marks)

1. Use the identity
   

    

   to show that
    
       ,
    
   where is a positive integer.  (2 marks)
   

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2. A club has members, with women and men.
   

    

   A group consisting of an even number of members is chosen, with the number of men equal to the number of women.
    
   Show, giving reasons, that the number of ways to do this is .  (2 marks)
   

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3. From the group chosen in part (ii), one of the men and one of the women are selected as leaders.
   

    

   Show, giving reasons, that the number of ways to choose the even number of people and then the leaders is
    
   

    

       .  (2 marks)
   

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4. The process is now reversed so that the leaders, one man and one woman, are chosen first. The rest of the group is then selected, still made up of an equal number of women and men.
   

    

   By considering this reversed process and using part (ii), find a simple expression for the sum in part (iii).  (2 marks)
   

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A fair standard die is rolled 50 times. Let be a random variable with binomial distribution that represents the number of times the face with a six on it appears uppermost.

1. Write down the expression for  , where  .  (1 mark)
   

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2. Show that  .  (2 marks)
   

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By using the fact that  , show that
 

.  (3 marks)

Show .  (1 mark)

Show .   (2 marks)

Using

show that

      (2 marks)