Permutation and Combination Questions Answers Formulas, Tips and Tricks

• 1. Permutation Formula and Facts

  
  

  1. Factorial Notation:
  
  Let n be a positive integer. Then, factorial n, denoted n! is defined as:
  
  n! = n(n - 1)(n - 2) ... 3.2.1
  
  Exmaple: 5! = 5*4*3*2*1 = 120
  Note: Please remember 0! = 1
  
  2. Permutation:
  The different arrangements of a given number of things by taking some or all at a time, are called permutations.
  
  Example:
  i. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
  
  ii. All permutations made with the letters a, b, c taking all at a time are:
  ( abc, acb, bac, bca, cab, cba)
  
  3. Number of Permutation:
  Number of all permutations of n things, taken r at a time, is given by:
  
  \begin{aligned}
  ^n P_r = n(n-1)(n-2).....(n-r+1) \\
  = \frac{n!}{(n-r)!} \\
  
  Example: \\
  ^4P_2 = 4 * 3 = 12 \\
  ^6P_2 = 6 * 5 = 30 \\
  
  \end{aligned}
  
  4. An Important Result:
  
  If there are n subjects of which p1 are alike of one kind. p2 are alike of another kind. p3 are alike of third kind and so on and pr are alike of rth kind,
  such that (p1 + p2 + ... pr) = n.
  
  Then, number of permutations of these n objects is =
  
  \begin{aligned}
  \frac{n!}{(p_1!)(p_2!)(p_3!)....(p_r!)}
  \end{aligned}

  • Comment on this formula

• 2. Combination formula and facts

  
  

  What is combination:
  Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
  
  Examples:
  
  1. Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
  
  Note: AB and BA represent the same selection.
  
  2. All the combinations formed by a, b, c taking ab, bc, ca.
  
  3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
  
  4. Various groups of 2 out of four persons A, B, C, D are:
  
  AB, AC, AD, BC, BD, CD.
  
  5. Note that ab ba are two different permutations but they represent the same combination.
  
  6. Number of Combinations:
  The number of all combinations of n things, taken r at a time is:
  \begin{aligned}
  ^n C_r = \frac{n!}{(r!)(n-r)!} \\
  = \frac{\text{n(n-1)(n-2).... to r factors}}{r!} \\
  \text{Note :} ^n C_n = 1 &, ^n C_0 = 1
  \end{aligned}

  • Comment on this formula
Permutation and Combination Questions Answers