“A permutation of a set of distinct objects is an ordered arrangement of these objects.”
Theorem: “If n is a positive integer and r is an integer with 1 ≤ r ≤ n, then there are P (n, r ) = n(n − 1)(n − 2) · · · (n − r + 1) r-permutations of a set with n distinct elements.”
Corollary: “If n and r are integers with 0 ≤ r ≤ n, then P(n,r) = n! / (n−r)!”
Definition “The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 ≤ r ≤ n, equals C(n,r) =n!/ r!(n−r)! ”
So, let’s start.
a) List all the 3-permutations of S
First, let’s calculate how many 3-permutations of S are there, so we don’t miss even one.
P(5,3) = 5!/(5-3!)=5!/2!=5*4*3*2!/2!=60.
So, let’s start:
123, 124, 125, 132, 134, 135, 142, 143, 145, 152, 153, 154,
213, 214, 215, 231, 234, 235, 241, 243, 245, 251, 253, 254
312, 314, 315, 321, 324, 325, 341, 342, 345, 351, 352, 354
…
b) List all the 3-combinations of S
C(5,3) = 5!/3!(5-3)!=5!/3!2!=5*4*3!/3!2!=10
In this case, the important thing to remember is in combinations, the order does not matter. So, 123 is the same as 231 and 321.
123, 124, 125, 134, 135, 145, 234, 235, 245, 345
Related exercises:
- List all the permutations of {a, b, c}.
- A professor writes 40 discrete mathematics true/false questions. Of the statements in these questions, 17 are true. If the questions can be positioned in any order, how many different answer keys are possible?
- How many different permutations are there of the set {a,b,c,d,e,f,g}?
- How many permutations of the letters ABCDEFG contain
- In how many ways can a set of five letters be selected from the English alphabet?
- How many bit strings of length 10 contain
- How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?
