Lecture 10

Permutations and Combinations
Permutations and combinations give us quick, algebraic methods of counting.
They are used in probability problems for two purposes:
1) to count the number of equally likely possible results for the classical approach to probability, and
2) to count the number of different arrangements of the same items to give a multiplying factor.
FACT 1: The number of distinct Permutations of n objects is “n factorial”, denoted by n !=n( n- 1 )…(2)(1)
FACT 2: The number of distinct Permutations of n objects, taken k at a time, is given by the ratio
P_r^n=(n !)/((n-k)!)=(n( n- 1 )…(2)(1))/((n-k)(n-k-1)…1)=n( n- 1 )…(n-k+1)
FACT 3: The number of distinct Combinations of n objects, taken k at a time, is given by the ratio C_r^n=(n !)/(k!(n-k)!)=(n( n- 1 )…(n-k+1))/k!
This quantity is usually written as C_r^n or (n¦k) and read "n choose k"
Question 3: Finally suppose that instead of portraits (“permutations”), we wish to form committees (“combinations”) of k = 3 people from the original n = 5. How many such distinct committees are possible