multipication principle (fundamental counting principle)
: suppose that an experiment E1 has n1 outcomes and, for each of these possible outcomes, an experiment E2 has n2 possible outcomes. Then the composite experiment E1E2 that consists of performing first E1 and then E2 has n1n2 possible outcomes.
permutation
: an ordered sequence or arragnement of the elements of a set
(usually assumes each element appears exactly once)
ex) there are 7 different locks, each with different key how many different ways can we match 1 key to each lock?
>> 7*6*5*4*3*2*1 = 7!ex) now suppose there are only 4 locks but 7 keys. how many ways to match a key to each lock?
>> 7*6*5*4 = 7!/(7-4)! = 7!/3!with replacement vs without replacement
1. with replacement
2. without replacement
- example
flipping 7 coins, 2 pennies, 3 nickels, 2 dimes
1. treating all 7 coins different -> 7! = 5040
2. without distinguishing in same coins(penny, nickel, dime) -> 7!/(2!3!2!) = 210
case of number 2
In general, the number of distinct permutations of n objects of k different types with n1,n1,...,nk (n1+n2+...+nk=n)
the number of distinct permutation is n!/(n1!n2!....nk!)
example of this:
how many 6 letter "words" can be formed with the letters from "pepper" -> 6!/(3!2!1!) = 60
combination
: a subset of elements from a set where the order is not important
- a partiton of a set into two groups in the subset or not in the subset
example) 12eggs and 4 are rotten. want to make 3 omelets.
1. how many omelets we can make without rotten eggs?
=> (8 3) = 8!/(3!(8-3)!) = 56
2. probability of the event above?
=> (8 3)/(12 3) = 56/220
Number of possible arrangement of size r from n objects
Question) what is the meaning of (n+r-1 r)?
