How many cells does the human body have?
Many. It depends on the person dimensions.
The only way I can see to tackle this problem is "being" a physicist. First of all, this is a quite difficult question, it could take a whole scientific paper to make the right estimation, likely it already exist out there.
Method 1: weighting to know
The first way is comparing the weight of a person with the weight of a single cell:
Where: N- number of cell; P- person's weight; C- cell's typical weight.
Method 2: measuring to know
The second way is by calculating:
Where: N- number of cells; PV- person's volume; CV- cell's volume.
I leave the math to you. A third way is comparing the surfaces instead of volumes in method 2, thus you need to make the calculation for each organ. All these methods will give you an approximation, quite bad, but it is a good starting point.
Assumptions . All cells are the same, e.g. bones, brain, hair...
The majority of the body mass, say 90%, is compacted within cells.
Take a person of 70 Kg, therefore
Limitations. this calculation is for healthy people, some medical condition such as obesity may alter the cell natural properties such as weight. Further, we took as granted that all cells are the same, wrong! but useful.
There is a high probability that this number is smaller than the real value, just in the brain we have
1. Molecular Cell Biology. 4th edition. See http://www.ncbi.nlm.nih.gov/books/NBK21473/. Accessed on 09 02 2016.
2. List of animals by number of neurons. https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons. Accessed on 09 02 2016.