Does a white blood count of 21, 000 indicate that the patient should be admitted to the hospital?

Not necessarily. I've done this to myself by swimming in cold water. Look at the big picture, what we know already, what kind of white cells they are, whether outpatient care is available, and even what resources may be available and what the patient wants.
Usually. The white count is not the main determinant of hospital admission.Other factors come into play such as what the cause is and the condition of the patient on presentation, availability of care givers for home treatment such as home nursing care and the decision of the patient. All these contribute to the decision making. Hospital admission risks hospital acquired infection.
No. It's just a number. For the most part we don't treat numbers. We treat the clinical findings. What is more important is why it's elevated, what are the findings on history and physical? How sick are they? A lab result by itself is merely a place to start. .