Show that x log x is O (x^2) but that x^2 is not O (x log x). In the Science Col

Question

Show that x log x is O (x^2) but that x^2 is not O (x log x). In the Science College of a university the faculty members are composed in 20 from mathematics, 15 from physics, 10 from chemistry and 8 from biology. How many committees with 8 faculty members can be selected considering that 3 must be from mathematics, 2 from physics and 1 from each of the other science departments? A professor writes 50 discrete mathematics true/false questions. Of the statements in these questions 28 are false. If the questions can be positioned in any order, how many different answer keys are possible? Use the binomial theorem to find the coefficient of the term with x^14 in the expansion of (x^3 - 1/x^2)^18. Assume that the probability a child is a boy is 0.51 and that the sexes of children born in a family are independent. What is the probability that the family of five children has at least one girl? What is the probability that a five-card poker hand contains cards of five different kinds? Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 99.9% of people with the disease test positive and only 0.02% who do not have the disease test positive. What is the probability that someone who test negative does not have the disease?

Explanation / Answer

2)

since there are 20 M ,15 P,10 C , 8 B faculty member

we have to select total 8 faculty member

since at least 3 from M , 2 from P,1 from C and 1 from B ,

so there are total 7 member we know from which department they will come,hence 1 extra member can be from any department

,so there are total 4 cases

a) 3M , 2P ,1C ,2B = 20C3*15C2*10C1*8C2 =1140*105*10*28

b)3M , 2P ,2C 1,B    = 20C3*15C2*10C2*8C1=1140*105*45*8

c)3M , 3P ,1C ,1B = 20C3*15C3*10C1*8C1=1140* 455 *10*8

d) 4M , 2P ,1C ,1B = 20C4*15C2*10C1*8C1=4845*105*10*8

we have to add the number of ways in each case

3)

since there are total 50 question and

there are 28 false ,and 22 true

so total number of answer key possible is the number of way we can arrange 28 F, 22 T

= 50! /(28!*22!)

5)

probability a child is boy = 0.51

probability a child is girl = 0.49

probability that the family of 5 children has at least one girl = 1 - probability that the family of 5 children has no girl

= 1 - (0.51)5  =0.965497

6)

probability that a five-hand poker hand ,in which all five are of different kind

= 13121110945

explanation - we have to select 5 kind out of 13 kind ,the number of ways = 13P5

for any kind it can of heart,club,diamond or spade ,so ther are 45 ways to do this

hence answer is 13C5 * 45 = 154440 * 1024 = 158146560

PS -please ask again question 1,4,7 on new post

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