Question

(iv) Check that the formula () = 21" holds for rows n = 0 to 5 in Pascal's triangle. (If it doesn't work for n = 4 or 5, go back and redo (iii)!) (v) Prove the formula of (iv) using (ii). (a) using (ii); (b) by proving that both sides of the formula represent the number of subsets of a set of n elements. For the left side use the addition rule for counting after partitioning the collection of all subsets according to size. And for the right side use the product rule for counting after identifying a subset A C {1,2,...,n} with the sequence of zeros and ones which is the indicator of A.

Answer 1

(m ) Read as in choose K" is the number of K. element subselo Subsels of an olan no element set n ! o where no nomy) (-2)---2:) . (mk)!K! Dascal theorem: for nik & Z, and k en the RoHS (me)! (m1)! with We coll plants ()+(mm) Torte y Craw; * (**)!K! K-1 - Kent)! pM+c)! KCK)! (AK) (0-)! (*)(N-K-1)!K! - K (04)! + (-)(-)! - - M-K)! KI mote K! =(n) n!

The identity of Binomial Crofficent let in and k.be positive integer with nak Suppose that I is a set Containing nt let elements a be an element in T. let S= T-{a} .There are (nel) subach oft containing k elements. However, a subnet OT with ok elements - either contain a together with (K-1) elements of sot Contains k elements of s and does not contain a Because there are you subset of (k-1) elements of s. there are ca subsets of k element of that contain a And there are (2) subset of k elements of that do not conloin a because there are .(") susets of k elements of s. Consequently ( ) = (2) + (A) - pascals Identity is the basis of geometric arrangent for bionomial Cefficents in a trangle 0 (0 : 1 =12 (3) (7) (3) 12 1 = 420= 4. Vandermonde's Idenlily let min. and & be non negalue entegers with or not exceeding either man. Then Cm?) - () () - In the result replace in with in the (7) = (n-1) $10K ) ( mada)