Download Combinations and Permutations: Understanding r-Permutations and r-Combinations - Prof. Tod and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity!
Lecture 14: Combinations and Permutations
- Associated reading: Rosen, Section 4.
- Homework: Section 4.3, problems 1,3,5,6,9,11,15,17,23,31,35,
r-Permutations: The number of ways to order r objects from a set of n objects is
P (n, r) = n(n − 1) · · · (n − r + 1).
If r = n (in other words, all n objects are ordered) then this n permutation is called a permutation for short.
Example 1. Suppose 3 people get on an elevator in a six-story building and each one gets off on a different floor. How many ways can this be done?
Example 2. How many injective functions are there from a set of 4 elements to a set of 6 elements? In general, how many injective functions are there from a set of m elements to a set of n elements?
r-Combinations: The number of ways to choose r objects from a set of n objects is
C(n, r) =
n(n − 1) · · · (n − r + 1) r!
n! (n − r)!r!
A more common way of writing C(n, r) is ( n r
)
C(n, r) is called a binomial coefficient since, in the expansion of (x + y)n, the coefficient of the xn−ryr^ term is C(n, r).
Example 5. What is the coefficient of the x^4 term in the expansion of (3x + 2)^6?
Example 6. The Birthday problem. What are the chances that two people in this classroom have the same birthday?
Theorem 1. Some combination identities
C(n, r) = C(n, n − r)
∑n k=0 C(n, k) = 2 n
∑n k=0(−1)
kC(n, k) = 0
- Pascal’s Identity: C(n, k) = C(n − 1 , k) + C(n − 1 , k − 1)
- Vandermonde’s Indentity: let k ≤ m, n. Then
C(m + n, k) =
∑^ k
r=
C(m, r)C(n, k − r)
Proof:
