My 5-year-old daughter keeps begging, throwing herself on the floor, and crying after I say "No." Her 3-year-old brother sees this and is starting to act the same way!
You are describing a 5 year old who is reacting to frustrations by falling apart, which is usually more typical in younger children. There must be a reason why she feels so hopeless and helpless at certain moments. Perhaps you already have clues about what types of situations set her off. Are there stresses in the family? Does her younger brother need a great deal of your time and attention? There may be certain pressures and demands put upon her that are exceeding her ability to cope.
Your daughter may be expressing distress and despair in the only way she can -- by acting like a baby. Naturally, we all have a tendency to loose patience with an older child who acts like a baby, but being annoyed with her won't help the situation. Anticipate problematic situations and head those off before they start so she won't get worked up. Consider offering a compromise before saying "No." Change how you respond and interact with her or provide distractions to side-step meltdowns. Or arrange some special mother-daughter or father-daughter time to help her handle frustrations better and have the ability to cope with stress in more mature ways.