Question

please provide the answers clearly

7. Suppose a Truc-False test has 20 questions (a) In how many ways may a student mark the test, if 6 questions are scored correctly and 14 incorrectly? (b) State which theorem from the text is applicable to solving part (a) (c) In how many ways may a student mark the test, if 10 questions are scored correctly and 10 incorrectly? 8. A baseball fan has a pair of tickets to 5 different home games of the Chicago Cubs. If the fan how many ways may he take one of them along to each of has four friends who like basebal the five games?

Answer 1

$(a)~We~can~select~6~questions~(which~are~answered~correctly)~out~of~20\\ ~in~\binom{20}{6}~ways.\\ Hence~total~of~ways~that~a~student~answers~6~correctly~out~of~20=\binom{20}{6}=\frac{(20)!}{6!(14)!}=38760.\\ (b)~The~number~of~ways~in~which~a~set~of~r~objects~can~be~formed~out~of~n~different~objects~is\\ \binom{n}{r}.\\ (c)~Required~no.~of~ways=\binom{20}{10}=\frac{(20)!}{(10)!(10)!}=184756\\ 8.~Since~the~fan~has~4~friends~and~he~has~a~pair~of~tickets~of~5~games~then\\ he~can~take~any~one~of~his~4~friends~along~to~each~of~the~five~games~in~5^4~ways.\\ Hence~total~no.~of~ways=5^4.$