The number of ways in which ten persons can shake hands with one another can be found by applying the formula for the number of combinations of n objects taken r at a time, which is nCr = n!/r!(n-r)!. Here, n = 10 and r = 2 (as each handshake involves two persons). Therefore, the number of ways is:

10C2 = 10!/(2!8!) = (10x9)/(2x1) = 45

Hence, the answer is option D, 45.