Ch. 2.3 Counting Sample Points. Cardinality of a Set

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1 Ch..3 Counting Smple Points CH 8 Crdinlity of Set Let S e set. If there re extly n distint elements in S, where n is nonnegtive integer, we sy S is finite set nd n is the rdinlity of S. The rdinlity of S is denoted y S or n(s). A set is sid to e infinite if it is not finite. Exmples 1-: 1. Let A = {1,, }. Then n(s) = 3.. Let S e the set of letters in the English lphet. Then n(s) =. 3. n({}) = 4. n( ) = 4. n({ }) =. n({ }) =. n({1}) = 7. n({1,,,1}) = 8. n({1,,1}) =. How mny smple points re in the rndom smple spe, S, when tossing 3 oins re tossed? We finlly egin to ount more omplited sets. In this setion we will utilize the following ounting tehniques Additive Rule Multiplition Rule Permuttions Comintions Additive Rule: n(a B) = n(a) + n(b) n(a B) If A nd B re disjoint, then n(a B) = n(a) + n(b) A B

2 CH Exmple 1: A survey of some UHD students on English nd Mth ourses determines tht of the students surveyed: 3 tke English this semester, 1 tke Mth nd tht 8 students re tking oth. How mny students were surveyed? Solution: Let E represent the set of UHD students who responded tht they re tking English. Let M represent the set of UHD students who responded tht they re tking Mth. Given: n(e) = 3 n(m) = 1 n( ) = 8 Question: n(e M) = Theorem 1 (The Multiplition rule): Suppose tht proedure n e roken down into two opertions. If there re n 1 wys to do the first opertion nd n wys to do the seond opertion fter the first opertion hs een done, then there re n 1 n wys to do the proedure. Note. The produt rule n e extended to more thn two tsks. Exmple 11: How mny smple points re in the smple spe when pir of die is thrown one? Solution: Proedure: throwing pir of die Opertion 1: the first die lnds, whih n e performed in wys. Opertion : the seond die lnds, whih n e performed in wys. By the multiplition rule, when throwing pir of die there re = 3 outomes. The smple spe n e desried s S = {(x,y) x nd y rnge from 1 to } Visulize y tree digrm:

3 CH 1 Exmple 1: A developer of new sudivision offers prospetive home uyers hoie of Tudor, rusti, olonil, nd trditionl exterior styling in rnd, two-story, nd split-level floor plns. In how mny wys n uyer order one of these homes? Tudor rusti olonil trdition l rnh two-story split-level rnh two-story split-level rnh two-story split-level rnh two-story split-level There re wys uyer n order one of these homes. Exmple 13: How mny four-digit numers n e formed from the digits, 1,,,, nd if eh digit n e used only one? Numer of possiilities Thousnds Hundreds Tens Ones Thousnds 1 Hundreds 1...

4 CH 11 Exmple 14: How mny even four-digit numers n e formed from the digits, 1,,,, nd if eh digit n e used only one? Tens Ones Thousnds 1 Hundreds 1 This doesn t work with the multiplition rule, so wht shll we try?

5 CH 1 Definition: A permuttion is n rrngement of ll or prt of set of ojets. Exmple 1: Consider the three letters, nd. The possile permuttions of ll three ojets re,,,. 1 st nd 3 rd Definition: Let n e positive integer. n ftoril is denoted n! is the numer n (n-1) (n-) (n-3) 1. Theorem 3 The numer of permuttions of n distint ojets is n!. Exmple 1: Consider the three letters,, nd d. The possile permuttions of two of the four ojets,,,. 1 st d nd d d d d d d d d d

6 CH 13 Theorem 4 The numer of permuttions of n distint tken r t time is n! n P r = ( n r)! Exmple 17: Three wrds (reserh, tehing nd servie) will e given one yer for lss of grdute students in sttistis deprtment. If eh student n reeive t most one wrd, how mny possile seletions re there? Exmple 18: A president nd tresurer re to e hosen from student lu onsisting of people. How mny different hoies of offiers re possile if ) there re no restritions. ) A will serve only if he is president ) B nd C will serve together or not t ll d) D nd E will not serve together

7 CH 14 Definition: Permuttions tht our y rrnging ojets in irle re lled irulr permuttions. Exmple 1: Consider the four people, Jun, nd Sm. How mny wys n they sit t irulr tle? Sm Jun Sme s Sm Jun Jun Sm Jun Sm Jun Sm Sm Sm Jun Jun Theorem The numer of permuttions of n distint ojets rrnged in irle is (n-1)!.

8 CH 1 Theorem The numer of distint permuttions of n things of whih n 1 re of one kind, n of seond kind, n k of k th kind is n! n! n! n 1 k! Exmple : In ollege footll trining session, the defensive oordintor needs to hve 1 plyers stnding in row. Among these 1 plyers, there re 1 freshmn, sophomore, 4 juniors nd 3 seniors, respetively. How mny different wys n they e rrnged in row if only their lss level will e distinguished? Let f 1 e the freshmn, sp 1 nd sp e the sophomores, j 1, j, j 3 nd j 4 the juniors nd sn 1, sn nd sn 3 the seniors. An exmple of the 1 plyers stnding in row 1) f sp sp j j j j sn sn sn whih is the sme s the following f 1 sp 1 sp j 1 j j 3 j 4 sn 1 sn sn 3 f 1 sp 1 sp j 1 j 3 j j 4 sn 1 sn sn 3 f 1 sp 1 sp j 1 j 4 j j 3 sn 1 sn sn 3 f 1 sp sp 1 j 1 j j 3 j 4 sn 1 sn sn 3 f 1 sp sp 1 j 1 j 3 j j 4 sn 1 sn sn 3 f 1 sp sp 1 j 1 j 4 j j 3 sn 1 sn sn 3 Aording to Theorem the nswer is 1!/[1!!4!3!]= 1.

9 CH 1 Theorem 7 The numer of rrngement of set of n ojets into r ells with n 1 elements in the first ell, n elements in the seond, nd so forth, is n n! = n1, n,..., nr n1! n! nr! where n 1 + n + + n r = n. Note: Theorem 7 is formliztion of Theorem. Also Note: Theorem 7 is one wy to motivte nd explin omintions s in Theorem 8. Theorem 8 The numer of omintions of n distint ojets tken r t time is n n n! = = r r, n r r!( n r)! Exmple 1: In how mny wys n 7 sientists e ssigned to one triple nd two doule hotel rooms? Note we would use Theorems 7 if the order of prts of the rrngement do not mtter nd the whole set is eing rrnged. If only susets of the whole set re eing rrnged then Theorem 8 should e tried. Exmple : A young oy sks his mother to get five gme-oy rtridges from his olletion of 1 rde nd sports gmes. How mny wys re there tht his mother will get 3 rde nd sports gmes, respetively?