How does Mormonism (the LDS Church) answer "the problem of evil"? My question this evening is:

How does Mormonism (the LDS Church) answer "the problem of evil": if God is all-powerful, all-knowing, and completely good and benevolent, why does evil exist in the world? (The premise is that a God with these attributes should know about the evil, be capable of stopping it, and be benevolent and good enough that he would necessarily act on that capability.)

I would appreciate it if answers would also explain whether the LDS view on God explains whether or not He knew humans would fall into sin. If so, why would He allow such a thing?
The problem with the Problem of Evil is that aside from the stated points, it also includes an unspoken premise that this life is all that matters.  If that premise is true, then the Problem of Evil exposes a very serious flaw in Christian theology: the very concept of the Christian God is self-contradictory, and therefore invalid.

Christianity, however, is fundamentally opposed to that concept, rejecting it in favor of what's known as "an eternal perspective": we believe that this life is a step along our eternal journey, that what truly matters is the afterlife, and that the most basic purpose of this life is to give us each a way to determine the state of our afterlife.

The Book of Mormon actually makes this explicit in various places.  For example, Alma 34: 32-34 states:


  32 For behold, this life is the time for men to prepare to meet God;
  yea, behold the day of this life is the day for men to perform their
  labors.
  
  33 And now, as I said unto you before, as ye have had so many
  witnesses, therefore, I beseech of you that ye do not procrastinate
  the day of your repentance until the end; for after this day of life,
  which is given us to prepare for eternity, behold, if we do not
  improve our time while in this life, then cometh the night of darkness
  wherein there can be no labor performed.
  
  34 Ye cannot say, when ye are brought to that awful crisis, that I
  will repent, that I will return to my God. Nay, ye cannot say this;
  for that same spirit which doth possess your bodies at the time that
  ye go out of this life, that same spirit will have power to possess
  your body in that eternal world.


This passage establishes that this life is "the time for men to prepare to meet God," and that a person's fundamental character will have been established by the point of death.  This is a recurring theme; the Book of Mormon refers to mortal life as "[day/days/state] of probation" in nine different places.

The prophet Lehi explains the basic concept in 2 Nephi 2:11-27:


  11 For it must needs be, that there is an opposition in all things. If not so, my firstborn in the wilderness, righteousness could not be brought to pass, neither wickedness, neither holiness nor misery, neither good nor bad. Wherefore, all things must needs be a compound in one; wherefore, if it should be one body it must needs remain as dead, having no life neither death, nor corruption nor incorruption, happiness nor misery, neither sense nor insensibility.
  
  12 Wherefore, it must needs have been created for a thing of naught; wherefore there would have been no purpose in the end of its creation. Wherefore, this thing must needs destroy the wisdom of God and his eternal purposes, and also the power, and the mercy, and the justice of God.
  
  13 And if ye shall say there is no law, ye shall also say there is no sin. If ye shall say there is no sin, ye shall also say there is no righteousness. And if there be no righteousness there be no happiness. And if there be no righteousness nor happiness there be no punishment nor misery. And if these things are not there is no God. And if there is no God we are not, neither the earth; for there could have been no creation of things, neither to act nor to be acted upon; wherefore, all things must have vanished away.
  
  14 And now, my sons, I speak unto you these things for your profit and learning; for there is a God, and he hath created all things, both the heavens and the earth, and all things that in them are, both things to act and things to be acted upon.
  
  15 And to bring about his eternal purposes in the end of man, after he had created our first parents, and the beasts of the field and the fowls of the air, and in fine, all things which are created, it must needs be that there was an opposition; even the forbidden fruit in opposition to the tree of life; the one being sweet and the other bitter.
  
  16 Wherefore, the Lord God gave unto man that he should act for himself. Wherefore, man could not act for himself save it should be that he was enticed by the one or the other.
  
  17 And I, Lehi, according to the things which I have read, must needs suppose that an angel of God, according to that which is written, had fallen from heaven; wherefore, he became a devil, having sought that which was evil before God.
  
  18 And because he had fallen from heaven, and had become miserable forever, he sought also the misery of all mankind. Wherefore, he said unto Eve, yea, even that old serpent, who is the devil, who is the father of all lies, wherefore he said: Partake of the forbidden fruit, and ye shall not die, but ye shall be as God, knowing good and evil.
  
  19 And after Adam and Eve had partaken of the forbidden fruit they were driven out of the garden of Eden, to till the earth.
  
  20 And they have brought forth children; yea, even the family of all the earth.
  
  21 And the days of the children of men were prolonged, according to the will of God, that they might repent while in the flesh; wherefore, their state became a state of probation, and their time was lengthened, according to the commandments which the Lord God gave unto the children of men. For he gave commandment that all men must repent; for he showed unto all men that they were lost, because of the transgression of their parents.
  
  22 And now, behold, if Adam had not transgressed he would not have fallen, but he would have remained in the garden of Eden. And all things which were created must have remained in the same state in which they were after they were created; and they must have remained forever, and had no end.
  
  23 And they would have had no children; wherefore they would have remained in a state of innocence, having no joy, for they knew no misery; doing no good, for they knew no sin.
  
  24 But behold, all things have been done in the wisdom of him who knoweth all things.
  
  25 Adam fell that men might be; and men are, that they might have joy.
  
  26 And the Messiah cometh in the fulness of time, that he may redeem the children of men from the fall. And because that they are redeemed from the fall they have become free forever, knowing good from evil; to act for themselves and not to be acted upon, save it be by the punishment of the law at the great and last day, according to the commandments which God hath given.
  
  27 Wherefore, men are free according to the flesh; and all things are given them which are expedient unto man. And they are free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil; for he seeketh that all men might be miserable like unto himself.


Important points found here:


There must be an opposition in all things, in order for meaningful distinctions between things to exist
Good and evil exist in opposition to each other
This life is a state of probation in which we are free to make meaningful choices between good and evil, and to repent of our sins when we choose evil
Satan entices mankind to choose evil, and Christ (the Messiah, the great Mediator) entices mankind to choose good
"Men are that they might have joy," which can only be meaningful if they also have an understanding of misery
Our moral choices have eternal consequences on the destiny of our souls


Therefore, if God were to do away with all evil, as the Problem of Evil suggests ought to be the case, there would be no meaningful moral choices to make, which would nullify the entire plan.