In class we studied the identity (r) combination (r) + (r+1) combination (r) +(r+2) combination (r) + ... + (n) combination (r) = (n+1) combination r+1 We also took a glimpse at (r) combination (0) + (r+1) combination (1) +(r+2) combination (2) +... +(n) combination (n-r) = (n+1) combination (n-r). We will now take a closer look at this second identity.

(a) Confirm that the second identity works for n=5, r=2 and for n=7, r=3.

(b) What is the relationship between the first and second identities?

(c) Prove the second identity above algebraically without using what you learned in Part b. (In other words, prove it without the help of the hockey stick identity we studied in class).

(d) Prove the second identity above with a block-walking argument.