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:
Hope this helps :)
Step-by-step explanation:
Median: 58
Lower Quartile: 55
Upper Quartile: 63
Minimum: 46
Maximum: 66
Answer:
$27.85
Step-by-step explanation:
I think what it means by outstanding charge is that there is something else he also needs to pay that he hasn't input yet. So, going off of that logic,
20+7.85 = 27.85, to find the amount he should pay for both charges to not create another bank fee.
Answer:
(x, y, z) = (0, 0, 3)
Step-by-step explanation:
The augmented matrix for the system is ...
![\left[\begin{array}{cccc}4&1&-2&-6\\2&-3&3&9\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D4%261%26-2%26-6%5C%5C2%26-3%263%269%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
Subtract 4 times the 3rd row from the first row.
![\left[\begin{array}{cccc}0&9&-2&-6\\2&-3&3&9\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%269%26-2%26-6%5C%5C2%26-3%263%269%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
Subtract 2 times the 3rd row from the second row.
![\left[\begin{array}{cccc}0&9&-2&-6\\0&1&3&9\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%269%26-2%26-6%5C%5C0%261%263%269%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
Subtract 9 times the 2nd row from the first row.
![\left[\begin{array}{cccc}0&0&-29&-87\\0&1&3&9\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%260%26-29%26-87%5C%5C0%261%263%269%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
Now, the first row can be divided by -29 to give ...
![\left[\begin{array}{cccc}0&0&1&3\\0&1&3&9\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%260%261%263%5C%5C0%261%263%269%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
You can subtract 3 times this first row from the second row to get ...
![\left[\begin{array}{cccc}0&0&1&3\\0&1&0&0\\1&-2&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%260%261%263%5C%5C0%261%260%260%5C%5C1%26-2%260%260%5Cend%7Barray%7D%5Cright%5D)
And add 2 times the second row to the third to get ...
![\left[\begin{array}{cccc}0&0&1&3\\0&1&0&0\\1&0&0&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D0%260%261%263%5C%5C0%261%260%260%5C%5C1%260%260%260%5Cend%7Barray%7D%5Cright%5D)
This matrix now tells you (x, y, z) = (0, 0, 3).
Answer:
V=216 ft cubed
Step-by-step explanation:
V=L*W*H
V=6*6*6
V=6*6=36
V=36*6
V=216
