A-committee-of-3-members-is-to-be-formed-from-8-members-Find-the-number-of-committees-that-can-be-formed-if-two-particular-club-members-cannot-both-be-in-a-committee-

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Question Number 138748 by physicstutes last updated on 17/Apr/21

$$\mathrm{A}\mathrm{committee}\mathrm{of}3\mathrm{members}\mathrm{is}\mathrm{to}\mathrm{be}\mathrm{formed}\mathrm{from}8\mathrm{members}.$$$$\mathrm{Find}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{committees}\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{formed}\mathrm{if}\mathrm{two}\mathrm{particular}$$$$\mathrm{club}\mathrm{members}\mathrm{cannot}\mathrm{both}\mathrm{be}\mathrm{in}\mathrm{a}\mathrm{committee}$$

Answered by mr W last updated on 17/Apr/21

$$C_3^8-C_1^6=50$$

Commented by physicstutes last updated on 17/Apr/21

$$\mathrm{sir}\mathrm{please}\mathrm{why}\mathrm{the}{}^6{\mathrm{C}}_1\mathrm{part}?$$

Commented by mr W last updated on 17/Apr/21

$$toselect3from8personsthereare$$$$totally{C}_3^{8}ways.$$$$iftwoparticularpersonsareinthe$$$$committee,toselectthethirdperson$$$$fromtherest6persons,thereare$$$$C_1^{6}ways.$$$$sincethesetwoparticularpersons$$$$shouldnotbebothinthecommittee,$$$$sowehaveonly{C}_3^8-C_1^{6}ways.$$

Commented by mr W last updated on 17/Apr/21

$$certainlyyoucanalsosolvelikethis:$$$$saythetwoparticularpersonsare$$$$AandB.$$$$1)bothAandBarenotincommittee$$$$weselectallthreemembersfromthe$$$$other6persons,thereare{C}_3^{6}ways.$$$$2)Aisincommitte$$$$weselectothertwomembersfrom$$$$theother6persons,thereare{C}_2^{6}ways.$$$$3)Bisincommitte$$$$weselectothertwomembersfrom$$$$theother6persons,thereare{C}_2^{6}ways.$$$$totally:thereare$$$$C_3^6+2\times C_2^6=50ways.$$

Commented by physicstutes last updated on 18/Apr/21

$$\mathrm{that}\mathrm{was}\mathrm{brilliant}\mathrm{sir}.\mathrm{Thanks}$$

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