Elizabeth Browning probably didn’t realize that she was really talking about mathematics when she penned her 43rd sonnet,How Do I Love Thee? This chapter provides a more comprehensive answer to this question than Browning was able to present in the remaining stanzas where she enumerates the ways she loves the veiled object of her sonnet. With the power of mathematics, equations are derived that provide a thorough enumeration, leaving no stone untouched. This is done through the simple expedient of selecting a set of items from a set. It is surprising, as when one falls in love, how fast innocent simplicity explodes into a tangled web of complexity. Perhaps this is what makes love stories, and mathematics, so enduringly interesting.

1. 1.

   For example, different colored balls aredistinguishable whereas electrons which have no discernible differences areindistinguishable.

2. 2.

   See the Appendix for a review of using recurrence to solve problems.

3. 3.

   The “r” superscript meanswith replacement rather than being a numeric index value.

4. 4.

   The restrictions to have non-negative arguments for\(p^r_{k,n}\) andp_k,n can be relaxed but will not be considered in this book.

5. 5.

   A smoothed plot of the frequency of birthdays ascends from a low around January until it reaches a peak in September. There is thus a greater chance that two or more people have a common birthday than what is calculated from equation (2.8).

6. 6.

   The value ofp_k,n and\(p^r_{k,n}\) soon swamps a computer’s floating point range for large argument values. This is the reason why the value is computed as the product of simple ratios.

7. 7.

   Convergence is quick. The value of\(\left | g_{13}/13! - 1/e \right |\) is about 10⁻¹¹.

8. 8.

   The product expansion (2.11) is used when computing the value of a binomial coefficient.

9. 9.

   This equation is termed Pascal’s equation after the French mathematician Blaise Pascal (1623–1662).

10. 10.

    Again, ther superscript meansreplacement rather than being an integer index.

Authors and Affiliations

1. (Home address), Beverly, MA, USA

   Randolph Nelson

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