Discrete Mathematics
WEN-CHING LIEN Department of Mathematics National Cheng Kung University
2008
1.4: Combinations with Repetition
Rule:
When we wish to select, with repetition, r of n distinct objects, we find (as in table 1.6) that we are considering all arrangement of r x’s and n−1 l ’s and that their number is
(n+r −1)!
r!(n−1)! =
n+r −1 r
.
Consequently, the number of combinations of n objects taken r at a time, with repetition, is C(n+r −1,r).
WEN-CHINGLIEN Discrete Mathematics
Example (1.29)
A donut shop offers 20 kinds of donuts. Assuming that there at least a dozen of each kind when we enter the shop, we can select a dozen donuts in
C(20+12−1,12) =C(31,12) =141,120,525 ways.
(Here n=20,r =12.)
Example (1.33)
Determine all integer solutions to the equation
x1+x2+x3+x4=7, where xi ≥0 for all 1≤i≤4.
Example (1.29)
A donut shop offers 20 kinds of donuts. Assuming that there at least a dozen of each kind when we enter the shop, we can select a dozen donuts in
C(20+12−1,12) =C(31,12) =141,120,525 ways.
(Here n=20,r =12.)
Example (1.33)
Determine all integer solutions to the equation
x1+x2+x3+x4=7, where xi ≥0 for all 1≤i≤4.
WEN-CHINGLIEN Discrete Mathematics
Example (1.29)
A donut shop offers 20 kinds of donuts. Assuming that there at least a dozen of each kind when we enter the shop, we can select a dozen donuts in
C(20+12−1,12) =C(31,12) =141,120,525 ways.
(Here n=20,r =12.)
Example (1.33)
Determine all integer solutions to the equation
x1+x2+x3+x4=7, where xi ≥0 for all 1≤i≤4.
Rule:
At this point it is crucial that we recognize the equivalence of the following:
a)The number of integer solutions of the equation
x1+x2+ · · ·xn=r, xi ≥0, 1≤i ≤n.
b)The number of selections, with repetition, of size r from a collection of size n.
c)The number of ways r identical objects can be distributed among n distinct containers.
WEN-CHINGLIEN Discrete Mathematics
Rule:
At this point it is crucial that we recognize the equivalence of the following:
a)The number of integer solutions of the equation
x1+x2+ · · ·xn=r, xi ≥0, 1≤i ≤n.
b)The number of selections, with repetition, of size r from a collection of size n.
c)The number of ways r identical objects can be distributed among n distinct containers.
Rule:
At this point it is crucial that we recognize the equivalence of the following:
a)The number of integer solutions of the equation
x1+x2+ · · ·xn=r, xi ≥0, 1≤i ≤n.
b)The number of selections, with repetition, of size r from a collection of size n.
c)The number of ways r identical objects can be distributed among n distinct containers.
WEN-CHINGLIEN Discrete Mathematics
Rule:
At this point it is crucial that we recognize the equivalence of the following:
a)The number of integer solutions of the equation
x1+x2+ · · ·xn=r, xi ≥0, 1≤i ≤n.
b)The number of selections, with repetition, of size r from a collection of size n.
c)The number of ways r identical objects can be distributed among n distinct containers.
Thank you.
WEN-CHINGLIEN Discrete Mathematics
