Mastering Mathematics Smartly
by Wee Wen Shih

This 6-page document highlights the essentials for H2 Statistics.

Additional points on permutations and combinations:

• Use Addition principle for mutually exclusive cases, e.g. 3-letter codes containing all E's, 3-letter codes containing 2 E's and another letter, 3-letter codes containing a single E and 2 other letters and 3-letter codes containing all other letters.
  
  
• Use Multiplication principle for tasks carried out in sequence, e.g. place the 5 men in a circle (i.e. task 1) and then place the 5 women in between men (i.e. task 2).
  
  
• Permutation is combination with order included.
  
  
• Number of ways to arrange n distinct objects in a row = n!.
  By definition, 0! = 1. 
  
  
• Number of ways to arrange n distinct objects in a circle = (n-1)!.
  In the case of a ring, we divide (n-1)! by 2.
  
  
• nPr = n! / { r! (n-r)! }, the number of ways to permutate (or arrange) r objects out of n objects.
  
  
• nCr = n! / (n-r)!, the number of ways to choose (or select) r objects out of n objects.
  Note that nPr = nCr . r! as we have mentioned in bullet 3.
  
  
• Suppose there are r1 identical objects, r2 identical objects, r3 identical objects among the n objects.
  The number of ways to arrange n objects = n! / { (r1)! . (r2)! . (r3)! }.
  
  
• Number of selections of varying sizes from n objects = nC0 + nC1 + ... + nCn = 2^n.
  
  
• In cases where we are given restrictions, we apply the restriction first. As an example, suppose we want to find the number of possible 5-digit numbers such that the result is an even number. We fix the last digit to be an even digit first. Then we consider the arrangement of the 4 digits.
  
  
• Consideration of complementary events is a good strategy at times, for its calcuation efficiency. As an example, suppose we want to find the number of ways in which a particular man and woman are to be separated. We consider the situation where they are together (i.e. the complementary event) and then we subtract it from the number of ways without any restriction (i.e. just arrange everybody), to give us the required answer.
  
  
• P&C can be mixed with probability and this is often tested. Essentially, we can the idea of probability as:
  
  probability = (number of ways of having a particular event) / (number of ways without any restriction)
  
  , where both numerator and denominator are treated as P&C calculations.