# Applied Discrete Structures by Alan Doerr, Kenneth Levasseur PDF

By Alan Doerr, Kenneth Levasseur

ISBN-10: 1365355942

ISBN-13: 9781365355943

Textbook from UMass Lowell, model 3.0

Applied Discrete constructions by way of Alan Doerr & Kenneth Levasseur is approved lower than an artistic Commons Attribution-NonCommercial-ShareAlike 3.0 usa License.

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4. COMBINATIONS AND THE BINOMIAL THEOREM 37 order of the three letters. By trial and error, abc, abd, acd, and bcd are the only listings possible. To repeat, we were looking for all three-element subsets of the set A. Order is not important in sets. The notation for choosing 3 elements from 4 is most commonly 43 or occasionally C(4, 3), either of which is read “4 choose 3” or the number of combinations for four objects taken three at a time. 3 (Binomial Coeﬃcient). Let n and k be nonnegative integers.

An be sets. 4 (Some generalized operations). If A1 = {0, 2, 3}, A2 = {1, 2, 3, 6}, and A3 = { 1, 0, 3, 9}, then 3 \ Ai = A1 \ A2 \ A3 = {3} i=1 and 3 [ Ai = A1 [ A2 [ A3 = { 1, 0, 1, 2, 3, 6, 9} i=1 With this notation it is quite easy to write lengthy expressions in a fairly compact form. For example, the statement A \ (B1 [ B2 [ · · · [ Bn ) = (A \ B1 ) [ (A \ B2 ) [ · · · [ (A \ Bn ) becomes A\ ⇣ ⌘ n n [ Bi = [ (A \ Bi ) i=1 i=1 20 CHAPTER 1. 1 Exercises A Exercises 1. Calculate the following series: a b P3 i=1 (2 + P1 2 i= 2 i c 3i) d 2.

How many ways can this be done if every student is to be in exactly one group? B Exercises 12. Explain in words why the following equalities are true based on number of subsets, and then verify the equalities using the formula for binomial coeﬃcients. 4. COMBINATIONS AND THE BINOMIAL THEOREM 41 13. There are ten points, P1 , P2 , . . , P10 on a plane, no three on the same line. a How many lines are determined by the points? b How many triangles are determined by the points? 14. How many ways can n persons be grouped into pairs when n is even?