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# 15b_00f_qs - 15B Weekly Quizzes Fall 2000 Q1 A family has 4...

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15B Weekly Quizzes Fall 2000 Q1. A family has 4 girls and 3 boys. (a) How many ways can they sit in a row? Ans: 7!. (b) How many ways can they sit in a row if boys and girls must alternate? Ans: 4! × 3! since the only possible seating pattern in GBGBGBG so we seat the boys (4!) and seat the girls (3!). Q2. How many ways can t teams each of size s be made from st people? The teams have no names or other distinguishing features. Three versions were given depending on student ID number: s =2 ,t =4; s =3 =3; s =4 . Ans: One could label the teams (count ordered teams). This will be t ! times the num- ber of teams since there are t ! ways to assign t labels to t diﬀerent sets of people. The number of labeled teams is ( st s,s,. ..,s ) and so the answer is ( st )! t !( s !) t . (One can do this with repeated binomial coeﬃcients instead of multinomial coeﬃcients: ( st s )( s ( t - 1) s ) ··· .) Q3. Let A , B and C B be sets. We make B A into a probability space by selecting functions from A to B uniformly at random. (a) What is the probability that a random f is an injection? Ans: Since there are altogher b a functions, each has probability 1 /b a . An injec- tion is an a -list without repetition from B , so there are ( b ) a of them. Thus the probability is ( b ) a /b a .

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15b_00f_qs - 15B Weekly Quizzes Fall 2000 Q1 A family has 4...

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