Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

356 B. Selected Solutions
doesn’t matter if you pick type 3 first and type 2 second, or the other
way around, just that you pick those two types).
(b) We want to consider injective functions from the set {most, second
most, second least, least} to the set of 10 cookie types. We want
injections because we cannot pick the same type of cookie to give
most and least of (for example).
(c) This is not a good problem to interpret as a function. The problem is
that the domain would have to be the 12 cookies you bake, but these
elements are indistinguishable (there is not a first cookie, second
cookie, etc.).
(d) The domain should be the 12 shapes, the codomain the 10 types of
cookies. Since we can use the same type for different shapes, we are
interested in counting all functions here.
(e) Here we insist that each type of cookie be given at least once, so
now we are asking for the number of surjections of those functions
counted in the previous part.
2.1.1.
2.1 Exercises
(a) Note that if we subtract 1 from each term, we get the square numbers.
Thus a n n 2 + 1.
(b) These look like the triangular numbers, only shifted by 1. We get:
a n n(n+1)
2
− 1.
(c) If you subtract 2 from each term, you get triangular numbers, only
starting with 6 instead of 1. So we must shift vertically and horizontally.
a n (n+2)(n+3)
2
+ 2.
(d) These seem to grow very quickly. Further, if we add 1 to each term,
we find the factorials, although starting with 2 instead of 1. This
gives, a n (n + 1)! − 1 (where n! 1 · 2 · 3 · · · n).
2.1.3.
(a) a 0 0, a 1 1, a 2 3, a 3 6 a 4 10. The sequence was described
by a closed formula. These are the triangular numbers. A recursive
definition is: a n a n−1 + n with a 0 0.
(b) This is a recursive definition. We continue a 2 2, a 3 3, a 4 4,
a 5 5, and so on. A closed formula is a n n.
(c) We have a 0 1, a 1 1, a 2 2, a 3 6, a 4 24, a 5 120, and so on.
The closed formula is a n n!.