Permutations and Combinations 1

Permutations and Combinations

• Factorial: The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). It is denoted as \( n! \).
  \( n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \)
  \( 0! = 1 \)
• Permutations: A permutation is a way of selecting and arranging objects in a particular order. The keyword is arranged.
  • The number of permutations of all \( n \) objects is given by:
    \( ^nP_n = n! \)
  • The number of permutations of \( r \) objects from \( n \) distinct objects is:
    \( ^nP_r = \frac{n!}{(n-r)!} \)
  • For permutations where there are \( p, q, r, \ldots \) of each type:
    \( \frac{n!}{p! \times q! \times r! \times \ldots} \)
  Example: How many ways can you arrange the letters in the word "BOOK"?
  The word "BOOK" has 4 letters where 'O' is repeated. The number of permutations is:

  \( \frac{4!}{2!} = \frac{24}{2} = 12 \)
• Combinations: A combination is a way of selecting objects without regard to the order. The keywords are chosen and selected.
  • The number of combinations of \( r \) objects from \( n \) distinct objects is:
    \( ^nC_r = \frac{n!}{r!(n-r)!} \)
  Example: How many ways can you choose 2 fruits from a basket of 5 different fruits?
  The number of combinations is:

  \( ^5C_2 = \frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = 10 \)

Quiz 1

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Quiz 2

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