2016 AMC 10A Prob 20
AMC problem solution and discussion.
Problem is of the "
Ball and Urn" type or "
Vending Machine" type. The AoPS folks often call this problem type "Bars and
Stars."
Partition problem sometimes refers to a different math problem.
This problem is analogous to finding the number of ways to construct the sum of positive integer exponents to add to
N:
A +
B + C +
D + E = N
Where (
A,B,C,
D,E) are the exponents of (a,b,c,d,e).
We can imagine the total sum as a line of N rocks that need to be divided into five piles whose individual sums constitute the exponents A,B,C,
D,E:
A ------- B ------------C-----------D------E
▀ ↑ ▀
. ▀ . ▀ ↑ ▀ . ▀ • • • ↑ ▀ . ▀ ↑ ▀
N
Rocks with 4 Partitions selected in (
N-1) available slots → C(N-1, 4)
Since all exponents are positive, this counting problem amount
...