Answer:
512
Step-by-step explanation:
Suppose we ask how many subsets of {1,2,3,4,5} add up to a number ≥8. The crucial idea is that we partition the set into two parts; these two parts are called complements of each other. Obviously, the sum of the two parts must add up to 15. Exactly one of those parts is therefore ≥8. There must be at least one such part, because of the pigeonhole principle (specifically, two 7's are sufficient only to add up to 14). And if one part has sum ≥8, the other part—its complement—must have sum ≤15−8=7
.
For instance, if I divide the set into parts {1,2,4}
and {3,5}, the first part adds up to 7, and its complement adds up to 8
.
Once one makes that observation, the rest of the proof is straightforward. There are 25=32
different subsets of this set (including itself and the empty set). For each one, either its sum, or its complement's sum (but not both), must be ≥8. Since exactly half of the subsets have sum ≥8, the number of such subsets is 32/2, or 16.
Answer:
100.00
Step-by-step explanation:
x=number of months until the costs are the same
50 + 20x = 25 + 30x
25 = 10x
x = 2.5 months
50 + 2.5(20) =$100.00
Answer:
Hello there! 10 square of 3
G = how many contests that Gertrude won
e = how many contests Elena won
Gertrude 4 more times than Elena
g = e + 4
They won a total of 18 contests
g + e = 18
