Code:
H = # houses
P = # plots
H >= P
1 house/plot
Houses distinguishable, Plots distinguishable
P(H,P) = H!/(H-P)! (# of permutations of H things taken P at a time)
For H=3, P=2, there are 6 ways:
A B
A C
B A
B C
C A
C B
Houses distinguishable, Plots indistinguishable
C(H,P) = H!/P!(H-P)! (# of combinations of H things taken P at a time)
For H=3, P=2, there are 3 ways:
A B
A C
B C
Houses indistinguishable
1 (each plot gets any house)
For H=3, P=2, there is 1 way:
H H
Any number of houses/plot
Houses distinguishable, Plots distinguishable
P^H (P choices for each house)
For H=3, P=2, there are 8 ways:
ABC {}
AB C
AC B
BC A
A BC
B AC
C AB
{} ABC
Houses distinguishable, Plots indistinguishable
This is the sum of the # of unique arrangements of houses for each of the ways to partition H into at most P partitions.
For H=3, P=2, there are 4 ways:
ABC {}
AB C
AC B
BC A
Houses indistinguishable, Plots distinguishable
C(H+P-1,H) = (H+P-1)!/H!(P-1)!
For H=3, P=2, there are 4 ways:
HHH {}
HH H
H HH
{} HHH
Houses indistinguishable, Plots indistinguishable
This is the number of ways to partition H into at most P partitions.
For H=3, P=2, there are 2 ways:
HHH {}
HH H
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