Total Number of all r-Combinations of n Objects. -- Suppose all the combinations of n objects taken one at a time (the number of the objects) are found, and then all the combinations taken two at a time, then all those three at a time, and so on until all those taken n at a time (all at once) are formed. How many combinations will there be altogether?

--J.E. Thompson, Algebra for the Practical Man (D. Van Nostrand, 1946)