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Math Help - Combinatorics / Inclusion/Exclusion Question

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    Combinatorics / Inclusion/Exclusion Question

    Hi everyone, first time posting on this forum and was hoping if someone would be willing to explain how to do the below question. Any help appreciated, thanks.

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    Re: Combinatorics / Inclusion/Exclusion Question

    Quote Originally Posted by floorplay View Post
    Hi everyone, first time posting on this forum and was hoping if someone would be willing to explain how to do the below question. Any help appreciated, thanks.
    The question is to find the number of solutions to \sum\limits_{k = 1}^5 {u_k }  = 24,\quad 4 \le u_1 \;\& \;u_2  < 7
    Let S_k be the event that u_1=k.
    Let T_k be the event that u_2=k.
    Let \|S_k\| be the number of solutions in S_k.

    The answer to the posed question is \sum\limits_{k = 4}^6 {\sum\limits_{j = 4}^6 {\left\| {S_k  \cap T_j } \right\|} },
    where \left\| {S_k  \cap T_j } \right\|=\binom{26-k-j}{2}.
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