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Math Help - Probabilities homework [stuck]

  1. #1
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    Probabilities homework [stuck]

    So I got this math homework and I'm having some problems resolving a few questions..
    .
    Is this correct? A74= 7!/(7-4)! = 7!/3!
    Qo2: Access codes to the schoolís network were given to students, teachers and technicians in a local high school: studentsí codes are composed of 4 different letters followed by 3 different numbers. Professorsí are formed of 2 letters (different or not) followed by 4 numbers (different or not). Techniciansí are made up of 1 letter and followed by 5 different numbers.

    1. How many different access code this high school can make?
    2. What is the probability that the symbols composing a professorsí code are all different?
    3. We know the letters and numbers composing a studentís code but do not know in which order they are disposed. What is the probability of finding the right combination the first time?


    I had a little trouble finding the answers to this questionÖbut hereís what I found:
    Students: L264= 26!/(26-4)!=26!/22!
    N103= 10!/(10-3)!=10!/7!
    LxN= 26!/22! + 10!/7! = (22x10)!/(7x26)! = 220!/182!
    Students have the possibility of 220!/182! Different access codes.

    Teachers: L=(262)= 26!/2!(26-2)!= 26!/2!24!=26x25/2=325
    N= (104)= 10!/4!(10-4)!= 10!/4!6! = 10x9x8x7/24= 210
    LxN= 325x210 = 68250
    Teachers have the possibility of 68250 different access codes.

    Technicians: L= 1/26
    N= A105= 10!/(10-5)! =10!/5! = 3628800/120
    1/26 x 3628800/120 = 15120/13
    Techs have the possibility of 15120/13 different access codes
    So the whole high school can make 220!/182! + 68250 + 15120/13 different access codesÖand Iím stuck here because I donít know how to sum all of this up =/ a little help please?
    b) L262= 26!/(26-2)! = 26!/24!
    N104= 10!/(10-4)! = 10!/6!

    • (26x6)!/(24x10)! = 156!/240!

    The probability that a teacherís access code is composed of different numbers and letters is 156!/240!

    Is anything above correct at all?
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  2. #2
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    Re: Probabilities homework [stuck]

    Hello, xluux!

    Access codes to a schoolís network were given to students, teachers and technicians.
    Studentsí codes are composed of 4 different letters followed by 3 different numbers.
    Professorsí are formed of 2 letters (different or not) followed by 4 numbers (different or not).
    Techniciansí are made up of 1 letter and followed by 5 different numbers.

    1. How many different access code this high school can make?

    \begin{array}{cccccc}\text{Students' codes:} & 26\cdot25\cdot24\cdot23\cdot10\cdot9\cdot8 &=& 258,336,000 \\ \text{Professors' codes:} & 26\cdot26\cdot10\cdot10\cdot10\cdot10 &=& \;\;\;6,760,000 \\ \text{Technicians' codes:} & 26\cdot10\cdot9\cdot8\cdot7\cdot6 &=& \;\;\;\;\;\;786,240 \\ \hline & \text{Total:} && 265,882,240\end{array}




    2.What is the probability that the symbols of a professorís code are all different?

    \text{All different: }\:26\cdot25\cdot10\cdot9\cdot8\cdot7 \:=\:3,276,000\text{ ways.}

    \text{Probability} \;=\;\frac{3,276,000}{67,760,000} \;=\;\frac{63}{130}




    3.We know the letters and numbers composing a studentís code
    but do not know in which order they are disposed.
    What is the probability of finding the right combination the first time?

    The student's four letters can be ordered in 4! = 24 ways.
    The student's three letters can be ordered in 3! = 6 ways.
    Hence, the student's code can have 24\cdot6 \,=\,144 possible orders.

    The probability of getting the right order is \frac{1}{144}
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