(a) Show that, for any positive integer n, = 2. )-() (:) - ()- =...

(a) Show that, for any positive integer n,

(b) Show that, for any positive integer n,

(c) Show that, for the 7th row of Pascal's triangle, the sum of the even-numbered elements equals the sum of the odd-numbered elements; that is,

(d) Use the result of parts (a) and (b) to show that, for any row of Pascal's triangle, the sum of the even-numbered elements equals the sum of the odd-numbered elements, and give that common sum for the nth row in terms of n.
(e) Suppose that S is a set of n elements. Use the result of part (d) to determine the number of subsets of S that have an even number of elements.
In the following triangular table, known as Pascal's triangle, the entries in the nth row are the binomial coefficients

Observe that each number (other than the ones) is the sum of the two numbers directly above it. For example, in the 5th row, the number 5 is the sum of the numbers 1 and 4 from the 4th row, and the number 10 is the sum of the numbers 4 and 6 from the 4th row. This fact is known as Pascal's formula. Namely, the formula says that

Transcribed Image Text:

п п п = 2". п )-() • (:) - ()- = 0. 0. +1