Mathematics

Problem

Consider the binomial coefficients

\binom{10}{k}

occurring in the expansion of

(1+1)^{10}

. Answer the following. a. Part 1: (i) Evaluate

S_1 = \displaystyle\sum_{k=0}^{6} \binom{10}{k}

. (A numerical value is expected.) b. Part 2: (ii) Using your answer to part (i), determine

S_2 = \displaystyle\sum_{k=4}^{10} \binom{10}{k}

. Explain your reasoning briefly. c. Part 3: (iii) Verify that

S_1+S_2

equals the appropriate power of 2 and comment on the result.