Prove a binomial expansion by induction

[stiffy]

This is an example done in class that I don't fully understand.; namely, the conclusion. Some explanation would be nice. The steps written down are steps I worked out. The teacher just showed were we had to end up. The rest I did. I may have made some typos...I hope not though. I didn't know how to show the n choose r so I just denoted it (n r) so its basically reading from left to right instead of up down. Sorry for that, it may be confusing. Nevertheless, I would appreciate some help.

Prove:

(x+y)^n=(n 0)x^n+(n 1)x^(n-1)*y^1+(n 2)x^(n-2)*y^2+...+(n n-1)x*y^(n-1)+(n n)y^n

This was assigned during class time and I worked it to...
(x+y)(x+y)^k=(k 0)x^k+(k 1)x^(k-1)*y^1+(k 2)x^(k-2)*y^2+...+(k k-1)x*y^(k-1)+(k k)y^k
=(x+y)[(k 0)x^k+(k 1)x^(k-1)*y^1+(k 2)x^(k-2)*y^2+...+(k k-1)x*y^(k-1)+(k k)y^k]

[(k 0)x^(k+1)+(k 1)x^(k-1)x^k*y^1+(k 2)x^(K-1)*y^2+...+(k k-1)x+(k k)y^k*x]
[(k 0)x^k*y+(k 1)x^(k-1)x^(k-1)*y^2+(k 2)x^(K-2)*y^3+...+(k k-1)x^(k-1)+(k k)y^(k+1)]

We see that (k 1) and (k 0) are the same, so we can say....(k 0)+(k 1)=(k+1 k)? I don't really understand the end of this proof. I would like to really understand the final statement "(k 0)+(k 1)=(k+1 k)." The teacher said to look at Pascal’s triangle if you don't get it...

This proof stemmed from a discussion about (n k)=(n-1 k) + (n-1 k-1)

Thanks,

Dan

 

[Deleted member 4993]

start with definition of (n, k)

(n,k) = n!/[k! (n-k)!] = [n(n-1)!]/[(n-k)k!(n-k-1)!] = n/(n-k) * (n-1)!/[k!(n-k-1)!]

(n-1, k) = (n-1)!/[k!(n-k-1)!]

And so on .....

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.