Homework 5 – Functions Part I and Part II

Homework 5 – Functions
In this homework assignment, you will be writing a few functions, to gain
experience using them and with parameter passing modes.

Part I: Combinatorics
The first part of this homework assignment will have you write a few basic functions
used widely in combinatorics.
1. Create a new project (called, for example, "homework05"), for this
assignment. Do not use spaces in the name. Follow the same process as
you did in the first assignment; if you need a refresher, please refer back to
Homework 1.
2. Open the "main.cpp" file in your project and delete everything in it.
3. Make sure that you include <iostream> and toss in a using namespace
std.
4. Write an empty main() function. Remember the int return type, and go
ahead and have it return 0 at the end so you don’t forget it later.
5. The first function that you will define is called factorial, which will
compute the mathematical factorial function. (Reminder: the factorial of a
non-negative n is written n! and is equal to 1 * 2 * … all the way up
to … * n. A special case is 0!, which by definition equals 1.)
The factorial function will take a single int parameter by value, and
return an int. Write the prototype (declaration) of this function above main.
Make sure you use the name factorial exactly, all lowercase.
Below main, write the definition of this function. Use a for loop to compute
the product of all of the integers from 1 up to and including the argument.
You can assume that the argument will be greater than or equal to 0.
If you get stuck, take a look at the sum function snippet that I did in class;
the two functions are very similar.
6. Test your function by writing a couple cout statements in main that call it.
For example, factorial(5)should return 120. Make sure that you try "edge
cases", like factorial(0), to make sure it works correctly. Compile and run
your code with those tests to see whether it works. If it doesn’t, go back and
try to figure out what you did wrong.
Be careful about testing it with large numbers; the factorial function grows
very rapidly, and will overflow a regular integer around factorial(13). We
could address this, but it’s not critical for this particular assignment, so move
on.

7. Next, we’ll compute the number of permutations of a set of objects by using
the factorial function we just wrote. (A permutation is a selection of
objects where order matters. For example, if a set contains the 4 elements A,
B, C, D, then ABC would be one permutation of 3 elements and BAC would be
a different one.) So, declare and define a function named permutations that
takes two ints by value, n and k, in that order. This function should return
the number of possible permutations of k objects selected from a set
of n total objects, computed by the following algorithm:
If k > n, the number of permutations is defined to be 0 (you can’t choose
more objects than the set contains). Otherwise, the formula for computing
the number of permutations of k objects selected from a set of n objects is:

Use your factorial function to write the formula above to compute the
number of permutations.
8. Test your permutations function in main just like you did
the factorial function. As an example, how many permutations are there
when you have a set of 4 objects and choose 2? Think about it and see if you
can come up with the answer on your own, then compare your answer to the
result generated by your function. Also try some edge cases, like
when k > n or when one of the values is 0.
9. Next, write a function named combinations that takes two ints by
value, n and k, in that order, and returns the number of
possible combinations of k objects selected from a set of n total objects. (A
combination is a selection of objects where order does not matter. For
example, if a set contains the 4 elements A, B, C, D, then you might only
count ABC but not BCA, CBA, and so on.)
If k > n, the number of combinations is defined to be 0 (you can’t choose
more objects than the set contains).Otherwise, the formula for computing
the number of combinations of k objects selected from a set of n objects is:

For the implementation of this function, don’t just call factorial three
times. Figure out how you would write this function by calling
the permutations function once and the factorial function once.

10. Test your combinations function. As an example, how many combinations
are there when you have a set of 4 objects and choose 2? Think about it and
see if you can come up with the answer on your own, then compare your
answer to the result generated by your function. Also try some boundary
cases, like when k > nor when one of the values is 0.

Part II: Passing by Reference
1. I want you to get some experience passing arguments by reference as well,
so let’s write one more function. This one will be called perms_and_combs,
and it computes both the number of permutations and the number of
combinations in a single function call.
2. So, declare and define a function named perms_and_combs that has
a void return type. It should take the four following arguments, in this
order: two ints, passed by value, which correspond to the n and k
values in Part I, followed by an int passed by reference which will be
used to store the number of permutations, followed by an int passed by
reference which will be used to store the number of combinations.
3. Inside the perms_and_combs function, call the permutations function and
store that result in the first reference parameter.
4. Then, again inside the perms_and_combs function, call
the combinations function and store that result in the second reference
parameter.
5. Test the new function inside your main, as before. Remember that you can
only pass variables as reference arguments to a function, so you’ll need to
declare two variables to hold the output of the perms_and_combs function
in order to get those results back.
6. Once you feel confident that your code is working the way it should, submit
it to Web-CAT. Note that when Web-CAT grades your submission, it will call
the four functions that you’ve written directly; your mainfunction will be
ignored, so focus on making sure that the functions produce the correct
results and don’t worry about formatting your output in any particular
way. Make sure the function names, and number of parameters is the
same as the description. Certain functions require you to use other
functions. The TAs will check that you have followed these instructions and
mark you down if haven’t. You will only receive half credit for a function if
it works but you haven’t followed directions.

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