I am working through some review questions for my mid term, and came across this problem:
I tried the problem, and my semi-complete solution looks just like the answer in the back, but I only got so far. Here is the answer in the back (with notes of me asking what is going on, and why):
*Above is how far I got, after this I have little understanding what is happening.*
*Okay, so where did the r come from on top on the left fraction? as-well as the r on the bottom. The right fraction suddenly is multiplied by "(n-r)"...why?*
Both fractions are multiplied by 1, which doesn't change them: the first one is multiplied by , and the second one by ...this is a
pretty standard trick in many parts of mathematics.
*this step also doesn't make sense to me. How does the bottom of the left fraction become that? I realize they have a common denominator now, but I do not understand their multiplication. On either side.*
For any natural , so on the left fraction's denominator , and , on the right one's denominator.
*Do not understand the addition here, at all. A webpage explaining something similar may clear this up. Will be searching after I finish typing this post*
They just added the fractions and factored out in the numerator!
*this makes sense, they added the r and -r together to get 0, so just left with the n on top*
*do not understand how the "(n-1)" disappears from the top here.*
Read the second note in blue
*This is how it is shown in the book. I assume the factorial still applies even when it is infront of the n. I understand this to be what C(n,r) looks like.
The ! sign in front of a number is either a typographic error, or else some non-standard notation they explain somewhere in the book, or somebody was high when printing the book, but it definitely is not the same as the ! sign AFTER the number, which is the standard, correct way to write it.
Hopefully the point of this thread makes sense to you. I am asking questions about an answer in the back, that I do not understand, and am looking for help with.
- Thanks in advanced!