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Math Help - counting and inclusion-exclusion problems

  1. #1
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    counting and inclusion-exclusion problems

    Hello, I'm new member, I found this website is very helpful. I need help with 2 review problems from my discrete class, please help me, thank you.

    1) Use inclusion-exclusion to find the number of 4-digit numbers with at least one 8 ( hint: let A_j = the set of all 4-digit numbers with an 8 in position i).

    here is my answer : 9 x 10 x10 x10 - 8 x 9 x 9 x 9 = 3168 , I'm not sure i get it right.

    2) Find the number of ways each situation below can occur:
    a) A ticket office has 5 reserved seat tickets and 8 general admission tickets to sell to 15 customers.
    b) Billy gets to pick 9 candy bars from the candy rack, which has Snickers,Mars,Payday, Mounds, and Crunch.

    I don't know how to do part a. For part b, I got P(9,5) = 15120
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  2. #2
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    Re: counting and inclusion-exclusion problems

    Quote Originally Posted by Tiome View Post
    1) Use inclusion-exclusion to find the number of 4-digit numbers with at least one 8 ( hint: let A_j = the set of all 4-digit numbers with an 8 in position i).

    here is my answer : 9 x 10 x10 x10 - 8 x 9 x 9 x 9 = 3168 , I'm not sure i get it right.
    Your solution is correct and is much simpler than the one that uses the inclusion-exclusion principle. Nevertheless, since the problem asks to use the principle: You need to find

    |A_1\cup A_2\cup A_3\cup A_4|=
    |A_1|+|A_2|+|A_3|+|A_4|-
    |A_1\cap A_2|-|A_1\cap A_3|-|A_1\cap A_4|-|A_2\cap A_3|-|A_2\cap A_4|-|A_3\cap A_4|+
    |A_1\cap A_2\cap A_3|+|A_1\cap A_2\cap A_4|+|A_1\cap A_3\cap A_4|+|A_2\cap A_3\cap A_4|-
    |A_1\cap A_2\cap A_3\cap A_4|

    It's pretty easy to find each of these terms.
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  3. #3
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    Re: counting and inclusion-exclusion problems

    Quote Originally Posted by Tiome View Post
    2) Find the number of ways each situation below can occur:
    a) A ticket office has 5 reserved seat tickets and 8 general admission tickets to sell to 15 customers.
    There will be 5 who get reserved, 8 who get general and 2 who get no ticket at all.
    How many ways can you rearrange the string RRRRRGGGGGGGGNN~?


    Quote Originally Posted by Tiome View Post
    2) Find the number of ways each situation below can occur:
    b) Billy gets to pick 9 candy bars from the candy rack, which has Snickers,Mars,Payday, Mounds, and Crunch.
    I don't know how to do part a. For part b, I got P(9,5) = 15120
    This is a multi-selection question. We are choosing nine from five kinds.
    This webpage explains the idea.
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  4. #4
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    Re: counting and inclusion-exclusion problems

    I got it, the link is very helpful, thank you so much.

    for 2a) i did 15!/(5!*8!) = 270270
    2b) C(9,5) = 9!/(5!*4!) = 126

    I have one more problem that i have no idea how to do it, if you can explain and help me , that would be great, thank you.
    Grid problem
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