# Thread: Combinatorics problem and some true and false.

1. ## 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)

2. ## Re: Combinatorics problem and some true and false.

Originally Posted by yummytree
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

3. ## 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: $\displaystyle \{A,A,B,B,C\}$ in some order.

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

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

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

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

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

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

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

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