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Math Help - Combinatorics problem and some true and false.

  1. #1
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    Combinatorics problem and some true and false.

    Hi All!!
    I was looking over some practice problems but I couldn't find the answers to them. So I was wondering if you guys can help me (Studying).
    (I'm going to apologize ahead of time because it might be a long post)

    How many 5-digit briefcase combinations contain


    1. Two pairs of distinct digits and 1 other distinct digit. (e.g 12215)
    I wasn't sure on which approach was correct.

    10 * 9 * 8 (because there are three distinct digits)

    or

    10C2 * 5C2* 3C2 * 8 (because you have to take into account how the doubles can be orientated)

    2. A pair and three other distinct digits. (e.g 27421)
    same issue as above

    10*9*8*7

    or

    (5C2*10) * 9 * 8 * 7


    I have found this question before but I couldn't get an explanation:

    2.How many ways are there to pick a collection of 12 coins from piles of pennies, nickels, dimes, quarters, and half-dollars? Base on the following condition:

    a. there are only 10 coins in each pile.
    16C4 - 5^2 because it's the total minus how many ways I can get from the 11th coin and the 12th coin. = 1795
    b. There are only 10 coins in each pile and the pick must have at least one penny and two nickels?
    1795 - 13C4(?)
    my logic is that it's because 12-3+4 C 4 but I'm not sure if I have to set it to 12 or 10.


    Lastly,

    1.Let B subset A and f : B subset Abe a 1-1 and onto, then B = A
    ( Sets A and B can be either finite or infinite.)
    T?

    2. A simple graph G with 13 vertices has 4 vertices of degree 4, 3 vertices of degree 3 and 6 vertices of degree 1, then G must be a tree.
    T?

    3.The spanning tree for any given graph is always unique.
    F?

    4.(A Union B subset A union C) then B subset C.
    T?

    5.Let A and B be nonempty sets and f: A->B be a function. Then if f(x n y) = f(x) n f(y) for
    all nonempty subsets X and Y of A, then f must be 1-1.
    T?


    Thank you! (If you help me I WILL LOVE YOU FOR THE REST OF MY LIFE)
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  2. #2
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    Re: Combinatorics problem and some true and false.

    Quote Originally Posted by yummytree View Post
    Hi All!!
    I was looking over some practice problems but I couldn't find the answers to them. So I was wondering if you guys can help me (Studying).
    (I'm going to apologize ahead of time because it might be a long post)

    How many 5-digit briefcase combinations contain


    10C2 * 5C2* 3C2 * 8 (because you have to take into account how the doubles can be orientated)
    I believe you had the right idea here but you left out a 9.

    pick a digit ->10
    pick 2 slots from 5 -> 5C2
    pick a digit from 9 -> 9
    pick 2 slots from 3 -> 3C2
    pick a digit from 8

    so 10*9*8*5C2*3C

    for 1 pair

    10 * 5C2 * 9 * 8 * 7 which is hopefully clear
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  3. #3
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    Lexington, MA (USA)
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    Re: Combinatorics problem and some true and false.

    Hello, yummytree!

    How many 5-digit briefcase combinations contain

    1. Two pairs of distinct digits and 1 other distinct digit. (e.g 12215)

    We have the digits: \{A,A,B,B,C\} in some order.

    There are 10 choices for the A, 9 choices for the B, 8 choices for the C.

    The digits can be arranged in: {5\choose2,2,1} \:=\:30 ways.

    Answer: . 10\cdot9\cdot8\cdot 30 \:=\:2160




    2. A pair and three other distinct digits. (e.g 27421)

    We have the digits: \{A,A,B,C,D\}

    There are: 10 choices for the A, 9 choices for the B,
    . . 8 choices for the C, 7 choices for the D.

    The digits can be arranged in: {5\choose 2,1,1,1} \:=\:60 ways.

    Answer: . 10\cdot9\cdot8\cdot7\cdot 60 \:=\:302,400
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