Chapter 5

COUNTING TECHNIQUESCOUNTING TECHNIQUESCHAPTER 5CHAPTER 5Ifonehastomakeapredictionregardingthe outcome of a certain activity, then the possibe outcomes re!uires the kno"edge of counting# In counting,thesimpestmethodisdoneby enumeratingathepossibeoutcomes#This method,ho"ever,istooaboriousandhence, mostanefficientmethod#Inthischapter,"e sha consider the most commony used methods ofcounting$thefundamentaprincipeof counting, permutation and combination#%&'(A)E'TA* PRI'CIP*E +% C+&'TI',Considerthenumbers-,.,/,and 0# 1uppose "e "ant to determine thetotat"o$digitnumberthat canbeformedifthesenumbers are combined#%irst, et us assume that no digit is to be repeated# Thusthepossibet"o$digitnumbersthatcanbe enumerated as foo"s2-. .- /- 0--/ ./ /. 0.-0 .0 /0 0/'oticethat"eareabetoe3haustathe possibiitiesthroughenumeration#Inthise3ampe, "e have -. possibe t"o$ digit numbers# Repetition is ao"ed2---. -/ -0.- .. ./ .0/- /. // /00- 0. 0/ 00Hence, "e have -4 possibe outcomes#If the first activity can be done in n-"ays andafterithasbeendone,thesecond activity can be done in n. "ays, then the totanumberof"aysin"hichactivities can be done is e!ua to n- 3 n.#E3ampe -2 Ho" many t"o$ digit numbers can be formed from the numbers -, ., /, 0 ifa# repetition is not ao"edb# repetition is ao"ed5a# n- 3 n.6 0 3 / 6 -. "aysb# n- 3 n. 6 0 3 0 6-4 "aysE3ampe .2Ho" many three$ digit numbers can be formed from the digits -, ., /, 0, 5 if any of the digits can be repeated5n- 6number of "ays of fiing thehundreds pacen. 6number of "ays of fiing the tenspacen/ 6number of "ays of fiing the unitpacen- 3 n. 3 n/ 65 3 5 3 5 6 -.5 "aysE3ampe /2Consider the numbers ., /, 5, and 7# If repetition is not ao"ed, ho" many three$ digit numbers can be formed such thata# they are a odds5b# they are a even5c# they are greater than 5885n- 6 be the number of "ays of fiing thehundreds pace#n. 6 number of "ays of fiing the tens pacen/ 6 numbers of "ays of fiing the unitpacen- 3 n. 3 n/ 6 . 3 / 3 / 6-9 "aysTo verify, the re!uired numbers are2./5 ./7 .5/ .57 .7/ .75/.5 /.7 /57 /75 5./ 5.757/ 5/7 7./ 7.5 7/5 75/n- 3 n. 3 n/ 6 . 3 / 3 - 64 "aysTo verify, the re!uired numbers are2/5. 5/. 7/./7. 57. 75.n- 3 n. 3 n/ 6 . 3 / 3 . 6-. "aysTo verify, the re!uired numbers are25./ 57. 7/. 5.7 57/ 75/7/5 5/. 7./ 75. 5/7 7.5PERMUTATIONThetermpermutationreferstothe arrangement of ob:ects "ith reference toorder#,ivenaset"ithnob:ects, then"ecantakerob:ectsfromthe set#Thetotanumberofpermutation ofnob:ectstakenratatimeis representedbythenotationnPrand can evauated using the formua2PERMUTATIONnPr 6; n r?