# Discrete Mathematics Elementary and Beyond by Axler, Gehring, Ribet

By Axler, Gehring, Ribet

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1 The Binomial Theorem In Chapter 1 we introduced the numbers nk and called them binomial coeﬃcients. It is time to explain this strange name: it comes from a very important formula in algebra involving them, which we discuss next. The issue is to compute powers of the simple algebraic expression (x+y). We start with small examples: (x + y)2 = x2 + 2xy + y 2 , (x + y)3 = (x + y) · (x + y)2 = (x + y) · (x2 + 2xy + y 2 ) = x3 + 3x2 y + 3xy 2 + y 3 , and continuing like this, (x + y)4 = (x + y) · (x + y)3 = x4 + 4x3 y + 6x2 y 2 + 4xy 3 + y 4 .

71828 . . ) This way we can deal with addition instead of multiplication, which is nice; but the terms we have to add up became much uglier! What do we know about these logarithms? 3). We have also drawn the line y = x − 1. We see that the function is below the line, and touches it at the point x = 1 (these facts can be proved by really elementary calculus). So we have ln x ≤ x − 1. 7) Can we say something about how good this upper bound is? From the ﬁgure we see that at least for values of x close to 1, the two graphs are 3 After all, the logarithm was invented in the seventeenth century by Buergi and Napier to make multiplication easier, by turning it into addition.

How many anagrams can you build from a given word? If you try to answer this question by playing around with the letters, you will realize that the question is badly posed; it is diﬃcult to draw the line between meaningful and nonmeaningful anagrams. For example, it could easily happen that A CROC BIT SIMON. And it may be true that Napoleon always wanted a TOMB IN CORSICA. It is questionable, but certainly grammatically correct, to assert that COB IS ROMANTIC. Some universities may have a course on MAC IN ROBOTICS.