Question: Write out several rows of Pascal's triangle. Circle the first 1 of any row. Go over one entry to the right and then one entry up and to the right, and circle that second entry (see Figure). Repeat this procedure until you run out of triangle. Then compute the sum of the entries you marked.
Do this procedure for several rows. What numbers do you get? Make a conjecture, then formulate your conjecture as an identity involving binomial coefficients. Finally, prove your conjecture is correct.