Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

If we allow repetitions in the permutations, then all of the previous theorems and
corollaries no longer apply. We have to start over !.
Theorem: The number of r-permutations of a set with n objects and repetition is
n r .
Proof: There are n ways to select an element of the set of all r positions in the r-
permutation. Using the product principle completes the proof.
Theorem: There are C(n+r-1,r) = C(n+r-1,n-1) r-combinations from a set with n
elements when repetition is allowed.
Example: How many solutions are there to x 1 +x 2 +x 3 = 9 for x i ,N? C(3+9-1,9) =
C(11,9) = C(11,2) = 55. Only when the constraints are placed on the x i can we
possibly find a unique solution.
Definition: The multinomial coefficient is C(n; n 1 , n 2 , …, n k ) = n!
k
! n i
!
.
i=1
93
Theorem: The number of different permutations of n objects, where there are n i ,
1;i;k, indistinguishable objects of type i, is C(n; n 1 , n 2 , …, n k ).
Theorem: The number of ways to distribute n distinguishable objects in k
distinguishable boxes so that n i objects are placed into box i, 1;i;k, is C(n; n 1 ,
n 2 , …, n k ).
Theorem: The number of ways to distribute n distinguishable objects in k
indistinguishable boxes so that n i objects are placed into box i, 1;i;k, is
k
j=11
j!
Multinomial Theorem: If n,N, then
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.
94