[This is a copy of an email I
sent about how to do this homework]
--------------------------
In
your answers to all the questions that ask you to give the number of some type
of objects (which is a majority of this homework), please give the answer first
of all in the "abstract" form, and only then (and this is entirely
optional), you can compute the numerical value of the answer . It's the
first form that we'll grade, not the latter!!!
What do I mean by "abstract" form? For example, if you are asked
what's the number of 4-digit numerical pins which contain digit 7, your answer
could be stated as:
10^4 - 9^4
[because the set of 4-digit pins which contain 7 is the set of all 4-digit
pins, whose cardinality is 10^4, minus the set of those 4-digit pins that do
not contain 7, and the cardinality of that latter set is 9^4, because it's a
set of 4-character strings each of which is chosen from a 9-element set of all
digits except 7.] (*)
Or it can be stated as follows (but that's more complicated):
4*9^3 + 6*9^2 + 4*9 + 1
[because the set of the pins that contain at least one 7 can also be
represented as a union of four disjoint sets: a set of all 4-digit pins
that contain exactly one 7, which contains 4*9^3 elements, the set of all
4-digit pins that contain exactly two 7's, which has 6*9^2 elements, a set of
4-digit pins that contain exactly three 7's, which has 4*9 elements, and a set
of pins that contain exactly four 7's, and this set contains a single pin...]
(**)
(Btw, you can see how much easier the problem becomes if you find a good way to
count: It's up to you to find an easy way to count, to make the problem
easier for yourself, but we'll not penalize a more difficult way if you choose
to follow it.)
If you give *either* of the above answers and leave it at this, you are going
to get full credit.
(We must therefore forewarn you that if you come up with some unusual way of
counting you might get no credit at first, and it'll be your job later on to
just convince us that your way of counting was correct, in which case we'll of
course change your grade.)
However, if *in addition* you compute this answer numerically, e.g. you answer:
10^4 - 9^4 = 10,000 - 6,561 = 3,439
That's of course fine, but we'll only grade the correctness of the "10^4 -
9^4" part.
(Actually, if you have two ways of counting something, computing it numerically
is an excellent way to check if your counting ways are
correct, so there's a point to doing that, but we don't require it and we'll
not grade it.)
On the other hand, if you just give us an answer 3,439 then we'll give you *no
points even though your answer is numerically correct*. Let me repeat
this: You get NO POINTS FOR
In problem 20 in section 5.1, you don’t have to compute expressions like floor(1000/7), floor(1000/11), floor(1000/77), etc.
[I’m using the word “floor” for a floor function here because I cannot find the
proper symbol for it in this text editor, but you should use the correct notation
for the floor and ceiling functions.] We'll accept answer(s) using
sums/differences/products of terms of this type, and providing the numerical
values of such expressions is optional. However, if you choose to just
provide numerical answers, it's fine *in this one exercise only*.
Also, unless we explicitly say so (and we have added a request for brief
justification to quite a few questions), it's up to you to provide an
explanation of the kind I gave in (*) or (**). It's a good idea that you
always do this anyway because, first of all, this is a way for you to see if
your logic is correct, and second, if you make a typo or a local error, there's
no way for the grader to understand what and why you are writing unless he/she
sees your verbal explanations.