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Lecture 6
1. Combinatorics
Recall the two basic principles of counting [combinatorics]:
First principle: m distinct garden forks plus n distinct fish forks equals m + n distinct forks.
Second principle: m distinct knives and n distinct forks equals mn dis- tinct ways of taking a knife and a fork.
2. Unordered Selection
Example 6.1. 6 dice are rolled. What is the probability that they all show different faces?
Ω =? |Ω| = 66. If A is the event in question, then |A| = 6 × 5 × 4 × 3 × 2 × 1.
Definition 6.2. If k is an integer! 1, then we define “k factorial” as the following integer:
k! = k · (k − 1 ) · (k − 2 ) · · · 2 · 1.
For consistency of future formulas, we define also
0! = 1.
Example 6.3. Five rolls of a fair die. What is P(A), where A is the event that all five show different faces? Note that |A| is equal to 6 [which face is left out] times 6 5. Thus,
P(A) = 6 ·^ 5! 65
3. Ordered Selection
Example 6.4. Two-card poker.
P(doubles) =
13 ×
( 4 × 3
2
( 52 × 51
2
Theorem 6.5. n objects are divided into r types. n 1 are of type 1; n 2 of type 2 ;
... ; nr are of type r. Thus, n = n 1 + · · · + nr. Objects of the same type are indistinguishable. The number of permutations is ( n n 1 ,... , n (^) r
= (^) n n! 1!^ · · ·^ n^ r!
Proof. Let N denote the number of permutations; we seek to find N. For every permtation in N there are n 1! · · · nr! permutations wherein all n objects are treated differently. Therefore, n 1! · · · n (^) r !N = n!. Solve to finish. "
Example 6.6. n people; choose r of them to form a “team.” The number of different teams is then n! r!(n − r)!
You have to choose r of type 1 (“put this one in the team”), and n − r of type 2 (“leave this one out of the team”).
Definition 6.7. We write the preceding count statistic as “n choose r,” and write it as (^) ( n r
n! r!(n − r)! =^
n! (n − r)!r! =
n n − r
Example 6.8. Roll 4 dice; let A denote the event that all faces are different. Then,
|A| =
