HI, please help with these exercises, please explain me step by step and please

Question

HI, please help with these exercises, please explain me step by step and please write with very good calligraphy. Thank you very much.

* The warehouse of a university received 25 printers, of which 10 are laser printers and 15 are models of inkjet. If 4 of those 25 are randomly selected for review by a particular technician, what is the probability that at least 3 of those selected are inkjet printers?

* Roberto invited 8 friends to his house, Juan and Pedro are two of them. If your friends arrive randomly and separately, what is the probability that Juan arrives just after Pedro?

* The President of the Republic has invited 10 senior Colombian executives to a meal at the Casa de Nariño. There are 10 posts in a row at a long table for those 10 guests. Upon arrival the executives are received and seated at random.
- 3 are from the Antioquia Syndicate
- 5 of the Santo Domingo Group
- 2 of the Ardila Lule Group
What is the probability that the members of each of the economic groups stay together?
What is the probability that the members of the Santo Domingo group stay together?

Explanation / Answer

Answer 1

Let Ij = {exactly j of the 6 selected are inkjet printers}, j = 3, 4, 5, 6.

This means

I3 = {exactly 3 of the 6 selected are inkjet printers}

I4 = {exactly 4 of the 6 selected are inkjet printers}

I5 = {exactly 5 of the 6 selected are inkjet printers}

I6 = {exactly 6 of the 6 selected are inkjet printers}

The experiment is selecting 6 printers from 25 (10 laser and 15 inkjet)

N (selecting 6 printers from 25) = 25C6 = 25!/[(6!)*(25-6)!] = 25!/(6!*19!) = 177100

N(I3) (selecting 3 printers from 15 inject printers and selecting 3 printers from 10 laser printer) = 15C3*10C3

N(I3) = [15!/(3!*12!)]*[10!/(3!*7!)] = 455*120 = 54600

P(I3) = N(I3)/N = 54600/177100 = 0.3083

N(I4) (selecting 4 printers from 15 inject printers and selecting 2 printers from 10 laser printer) = 15C4*10C2

N(I4) = [15!/(4!*11!)]*[10!/(2!*8!)] = 1365*45 = 61425

P(I4) = N(I4)/N = 61425/177100 = 0.3468

N(I5) (selecting 5 printers from 15 inject printers and selecting 1 printers from 10 laser printer) = 15C5*10C1

N(I5) = [15!/(5!*10!)]*[10!/(1!*9!)] = 3003*10 = 30030

P(I5) = N(I5)/N = 30030/177100 = 0.1696

N(I6) (selecting 6 printers from 15 inject printers and selecting 0 printers from 10 laser printer) = 15C6*10C0

N(I6) = [15!/(6!*9!)]*[10!/(0!*10!)] = 5005*1 = 5005

P(I5) = N(I5)/N = 5005/177100 = 0.0283

P(at least 3 inkjet printers are selected) = P(I3?I4?I5?I6) = P(I3) + P(I4) + P(I5)+ P(I6) = P(I3) + P(I4) + P(I5)+ P(I6)

P(at least 3 inkjet printers are selected) = 0.3083+0.3468+0.1696+0.0283

P(at least 3 inkjet printers are selected) = 0.8530

Answer 2

N (Arranging 10 people in 10 posts in a row) = 10P10 = 10!/(10-10)! = 10! = 3628800

Consider each group of people as one entity. Therefore, number of ways of arranging 3 entities = 3P3 = 3! = 6

As there are 3 members in Antioquia Syndicate. They can be arrange within themselves in 3! (6) ways  

As there are 5 members in Santo Domingo Group. They can be arrange within themselves in 5! (120) ways  

As there are 2 members in Lule Group. They can be arrange within themselves in 2! (2) ways

Number of ways in which members of each of the economic groups stay together = 3!*3!*5!*2! = 6*6*120*2 = 8640

P(that the members of each of the economic groups stay together)

= N (Arranging 10 people in 10 posts in a row) / Number of ways in which members of each of the economic groups stay together

P(that the members of each of the economic groups stay together) = 8640/3268800 = 0.0026

Now, consider people in Santo Domingo Group as one entity. So total number people becomes 6 [3(Antioquia Syndicate) +2(Ardila Lule Group)+1(Santo Domingo Group)]

So, these can be arranged in 6! (720) ways

As there are 5 members in Santo Domingo Group. They can be arrange within themselves in 5! (120) ways  

Number of ways in which members Santo Domingo Group stay together = 720*120 = 86400

P(the members of the Santo Domingo group stay together) = 86400/3268800 = 0.0264

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