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Math Help - 4 Questions

  1. #1
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    Lightbulb 4 Questions

    Question 1:
    Find the number of ways in which five balls of different color can arrange in a row.


    Question 2:
    A teacher has to select 5 boys from section A, 4 boys from section B and 3 boys from section C for a competition, where section A has total 8 boys, section B has total 7 boys and section C has total 6 boys. How many choices does the teacher have?


    Question 3:
    A variable name in a programming language must start with an alphabet or characters _, next letters can be an alphabet or number. Maximum size for the variable name is 5 characters. How many different variable names are possible?


    Question 4:
    In how many ways can nine students be partitioned into three teams containing four, three, and two students respectively?
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  2. #2
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    These are the simplest possible counting problems concerning queues.
    What have you done on any of them? Where is the difficulty?
    Tell us what you do not understand.
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  3. #3
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    I got this assignment, I know that the answers are very simple, but I wanted to confirm that these are really so simple that even the answer of the first question is of one line?
    5 x 4 x 3 x 2 x 1 so 120 is it fine?
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  4. #4
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    Right ! It's the permutation of 5 .
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    It's been a while since these questions were asked and I'm really bored, so I'll take a stab at one of the questions. Hopefully i don't embarrass myself too badly.
    Quote Originally Posted by aamirpk4 View Post
    Question 2:
    A teacher has to select 5 boys from section A, 4 boys from section B and 3 boys from section C for a competition, where section A has total 8 boys, section B has total 7 boys and section C has total 6 boys. How many choices does the teacher have?
    The teacher has {8 \choose 5} + {7 \choose 4} + {6 \choose 3} choices
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  6. #6
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    Hello, aamirpk4!

    4)In how many ways can nine students be partitioned into three teams
    containing four, three, and two students, respectively?
    Answer: . {9\choose4,3,2} \:=\:\frac{9!}{4!\,3!\,2!} \:=\:1260

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  7. #7
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    Quote Originally Posted by aamirpk4 View Post
    Question 3:
    A variable name in a programming language must start with an alphabet or characters _, next letters can be an alphabet or number. Maximum size for the variable name is 5 characters. How many different variable names are possible?
    +the First case: the first letter is alphabet .
    -The first letter has 26 choices
    -The 2sd letter has 36-1 choices (26 alphabetic characters and 10 numbers)
    -The 3rd letter has 36-2 choices.
    -...
    -...
    So , we have :26 + 26*35 + 26*35*34 + 26*35*34*33 + 26*35*34*33*32 =: S1 choices for this case.

    +the Second case: the first letter is "_" character.
    -The first has 1 choice.
    -The 2sd letter has 36 choices.
    -The 3rd letter has 36-1 choices.
    -...
    So , we have :36 + 36*35 + 36*35*34 + 36*35*34*33 =: S2
    (considering that "_" is not a variable name)

    The final result is S1+S2 .
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  8. #8
    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by Jhevon View Post

    The teacher has {8 \choose 5} + {7 \choose 4} + {6 \choose 3} choices
    I might be wrong, but should the combinations be multiplied instead of added?
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  9. #9
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    Quote Originally Posted by DivideBy0 View Post
    I might be wrong, but should the combinations be multiplied instead of added?
    Not when you divide the problem into sub-cases.

    For example, say the problem goes something like this "...find how many combinations for at least 3 students..."

    You can simplify the problem by finding:
    exactly 1 student
    exactly 2 students
    exactly 3 students

    And then add up the combinations in all the sub-cases.
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  10. #10
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    Quote Originally Posted by DivideBy0 View Post
    I might be wrong, but should the combinations be multiplied instead of added?
    Question 2:
    A teacher has to select 5 boys from section A, 4 boys from section B and 3 boys from section C for a competition, where section A has total 8 boys, section B has total 7 boys and section C has total 6 boys. How many choices does the teacher have?

    Yes you are correct we must multiply and not add to answer this partiucular question.
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  11. #11
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    Quote Originally Posted by Soroban View Post
    Hello, aamirpk4!

    Answer: . {9\choose4,3,2} \:=\:\frac{9!}{4!\,3!\,2!} \:=\:1260

    Hi Soronban.

    Another way :
    We choose 4 students from 9 students : {9\choose4}
    After that choosing 3 students from 5 students : 5\choose3
    And the final , choosing 2 from 2 : 2\choose2
    So , the result is : {9\choose4}* 5\choose3* 2\choose2 = 1260.

    @ Soronban : Would you like to tell how we receive the function that you gave ? I mean that what way you think about the question 4?
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  12. #12
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by DivideBy0 View Post
    I might be wrong, but should the combinations be multiplied instead of added?
    Quote Originally Posted by Plato View Post
    Question 2:
    A teacher has to select 5 boys from section A, 4 boys from section B and 3 boys from section C for a competition, where section A has total 8 boys, section B has total 7 boys and section C has total 6 boys. How many choices does the teacher have?

    Yes you are correct we must multiply and not add to answer this partiucular question.
    Indeed!

    looking back at this, i have no idea why i decided to add...
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