Question

Question 7: Consider 10 male students M1, M2,... , Mo and 7 female students F1, F2,... , F Assume these 17 students are arranged on a horizontal line such that no two female students are standing next to each other. We are interested in the number of such arrangements, where the order of the students matters Explain what is wrong with the following argument We are going to use the Product Rule: » Task 1: Arrange the 7 females on a line. There are 7! ways to do this . Task 2: Choose 6 males. There are0) ways to do this . Task 3: Place the 6 males chosen in Task 2 in the 6 "gaps" between the females. There are 6! ways to do this . Task 4: At this moment, we have arranged 13 students on a line. We are left with 4 males that have to be placed. Task 4.1: Place one male. There are 14 ways to do this Task 4.2: Place one male. There are 15 ways to do this Task 4.3: Place one male. There are 16 ways to do this - Task 4.4: Place one male. There are 17 ways to do this By the Product Rule, the total number of ways to arrange the 17 students is equal to 10 6 7! 6! 14. 15 -16-17 43,528, 181, 760, 000 Question 8: Let n 2 2 be an integer and consider the set S-11,2,... ,n]. An ordered triple (A, x, y) is called awesome, if (i) AC S, (ii) x A, and (iii) y A. . Let k be an integer with 1 < k < n. Determine the number of awesome triples (A,x,y) with A k Prove that the number of awesome triples (A, r, y) with y is equal to ● Determine the number of awesome triples (A,a, y) with xメy . Use the above results to prove that rl

Answer 1

The peoblem in the given ossange ment is that a (ot of aeongements Nous suppose in task too the choosen males ase Which ase placed as be(οω Nou Ge can place ma at any of the lu,plares soppose tn a rng and mio at the end. Then my assa ement : NOO Soppose in some othe8 A砌。gement , the chosen 6 notes ase ond ue plare thenn in the places as fotioue Now suppose Mi in the hext Position to place ond if on keep bim at any of the ty positions Supposei keep i at second position and M8, and Mo at last e get: Gse can see that the absange ment ase same bot abe counted as diffesent in the asgoment Ip this u can obsesre o a tot of axsongements ae counted mose thon ooce The cossect ato do t is: Prssange tbe to maies on the line which can be done in Nod e bave t ptaces to keep omen (te a gap be tcieen ue cin chose 6 Ptaces ost of these 11 and aonge aus women the roles if Staat and end position) ơs in this scenoso hexe whic'h can be dope in