Consider the set Vk of all possible k-component vectors x with elements x1, . . . , xk, each of which is either 0 or 1. (a) How many vectors are there in Vk? (b) Suppose you have a vector y ? Vkk1. Why is there exactly one vector x ? Vk whose first kk 1 elements are the elements of y and the sum of whose elements is an even number? (c) Use the result of part (b) to determine the number of vectors in Vk having the property that the sum of their elements is an even number. (d) Suppose now that k is even. By counting certain types of {0, 1} vectors, evaluate in closedform: k0 + k2 + k4 + k6 + + kk.

2. Consider the set Vk of all possible k-component vectors x with elements x1, ..., Ck, each of which is either 0 or 1. (a) How many vectors are there in Vk? (b) Suppose you have a vector y E Vk-1. Why is there exactly one vector x E Vk whose first k - 1 elements are the elements of y and the sum of whose elements is an even number? (c) Use the result of part (b) to determine the number of vectors in V having the property that the sum of their elements is an even number. (d) Suppose now that k is even. By counting certain types of {0, 1} vectors, evaluate in closed- form: + +