How many different solutions of non-negative integers x1, x2, ..., xk are there to
x1+x2+x3+...+xk = n ?
Answer: C(n+k-1, k-1)
Some people have indicated to me that this is 'obvious' through their forum
postings. But I would like you to explain to me why it is true from the
I will explain in class tonight one way of showing this is true through
an example. You could use induction or some other argument. I don't care
how, I am just looking for an explanation which is clear and simple as possible.
Please stick to:
1) The addition and multiplication principles
2) C(n,r) = the number of ways of choosing r elements from a set n distinct elements
3) P(n,r) = the number of ways of ordering r elements from a list of n distinct elements
I will use the same criterion that I use to evaluate the forum questions, namely I
am looking for clear, complete and short explanations. The 'short' criterion is
is important so please do not have a solution which goes on for more than a page
(because I won't be able to follow it and then it breaks the 'clear' part too).
Please do not work together on this assignment. I would like to see original compositions.
When you write anything using the words that do not clearly have anything to do
with the problem like 'choose' 'list' 'spaces' please make it clear what you mean.
Somehow you must explain how you transition from 'numbers of solutions to an equation' to
an actual value and so you must make it clear to someone who starts reading a
question about rearranging letters how you are applying the tools you have available
to you to arrive at a number value.
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