COMP 170 L2 Page 26 The Binomial Theorem l We are concerned with n What is the theorem true?
COMP 170 L2 Page 27 Examples l Monomial terms: n Lists of length two, each element can either be x or y. l How many monomial terms with one y (and hence one x) ? n = number of ways to choose 1 place among 2 places n That is the coefficient for the term l Similarly n Coefficient for = number of lists having 0 place for y = n Coefficient for = number of lists having 2 places for y = l So
COMP 170 L2 Page 28 Examples l Coefficient for n = number of ways to choose 2 places for 3 places. l Coefficient for n = number of ways to choose i places from 3 places
COMP 170 L2 Page 29 Proof of the Binomial Theorem l Coefficient of n = number of lists having y in k places n =number of ways to choose k places from n places n=n=
COMP 170 L2 Page 30 Applications of the Binomial Theorem
COMP 170 L2 Page 31 Applications of the Binomial Theorem
COMP 170 L2 Page 32 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients