# combinatorics

1. ### Pigeonhole Principle

Hello to all I am stuck with a question that goes like 410 letters are distributed in 50 apartments then which of the following are true S1 - Some apartment received atleast 9 letters S2 - Some apartment received atmost 8 letters S3 - Some apartment received atleast 10 letters S4 - Some...
2. ### Ordered and Unordered partitions

Hello This is my first post. I am unable to grasp the concept of ordered and unordered partitions in Combinatorics. 1) How many ways can 10 persons be divided into 3 teams such that Team-1 contains 3, Team-2 contains 2 and Team-3 contains 5 members respectively? The answer to this is...
3. ### combinatoric question

A frog in on a integer n_{0} on day 0 and every day performs a jump in the same direction and lenght, so on day k it is in n_{0}+ks, where s is the lenght of the jump. A blind eagle wants to eat it, but it's doesn't know n_{0} and s. It can only come down on a integer every day. Exist a strategy...
4. ### Proving an identity with unsigned sterling numbers of first kind

How do I prove c(n+1, m+1) = \sum_{k=0}^{n} c(n, k) \binom{k}{m}
5. ### Polygon types

Triangles can have 1 obtuse angle, 1 right angle, and 2-3 acute angles. The angles are integers between 0 and 180 but not including 0 and 180. How many triangles are there as far as angles and how many of them are obtuse, how many of them are right, and how many of them are acute? You...
6. ### combinatorics problem

Prove that total number of possibilities for x1,x2,x3,...,xr∈N-{0} such that x1+x2+x3+...+xr=n are (n-1)C(r-1)
7. ### 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...
8. ### Combinatorics

Hi, I uploaded two statements (I guess), and I didn't understand them , can someone help me understand why do they do k times Q(n-1,k-1) or k times Q(n-1,k)??? I don't understand the k times part.. And the lecturer wrote to the properties of Q, Q(n,1)=1.... shouldn't that be equal to n? Q is...
9. ### Counting subsets of disjoint sets

The set, A, contains numbers: \{1,2,\text{...},n\}. The set, B, contains any subset of A with cardinality 2: \{\{1,2\},\{1,3\},\text{...},\{n-1,n\}\} C_1,C_2,C_3 are some pairwise disjoint sets that satisfy C_1\cup C_2\cup C_3=B. The set, D, contains any subset of B with cardinality i, whose...
10. ### Combinatorics Pigeonhole problem

Hello to all! So i have to do this problem: In the course of an year of 365 days Peter solves combinatorics problems. Each day he solves at least 1 problem, but no more than 500 for the year. Prove that for the year there exists an interval of consecutive days in which he had solved exactly...
11. ### Combinatorics - choosing exactly k pairs from n

Hi. I have the following combinatoric problem (well it's actually a probability problem that need to be resolved using combinatorics): There are n pairs of shoes in the closet. 2m shoes are chosen from it randomly. (m<n) find the probability to get exactly k pairs. so this is what I'm...
12. ### Combinatorics Question

6 people each own a desk each. The company owner wants to swap them round, so at most one of them stays where they were before. How many arrangements are possible? I've tried 264, which was the apparent sum of the arrangements where everybody changes position plus the arrangements with one in...
13. ### Combinatorics - HELP! ( FCP)

Hi guys here is the question . An area code is the first 3 digits in a phone number and indicates the location of either the province or the city. In Canada, the following area codes exist: Manitoba 204 Saskatchewan 306 Québec (Québec City) 418 Montreal 514 Newfoundland 709 Québec 450...
14. ### Binary Sequence combinatorics

The problem simply states that: show that there cannot be 171 binary sequences such that each differ in at least four digits. So far I've deducted that there should be some pigeonhole principle in here (I'm assuming, at least). So far I've written down that we should take the 2^12 different...
15. ### Combinatorics Induction Proof

imgur: the simple image sharer Think I have parts 1 and 2, struggling with the induction :S Any help greatly appreciated!
16. ### Combinatorics question - 10 teams

Hi i want to ask you if you could help with this problem : I have 10 teams and 9 days... each day one team must play one match with another team (not same as already played before/diferent team). How will look table of matches ? I mean can you write each day which teams will play together ? I...
17. ### Combinatorics question

Hello!, My question is A hockey team has played 10 games and has a record of 5 wins, 3 losses and 2 ties. In how many ways could this have happened if after the first 4 games the team's record was 3 wins and a loss. This is what i thought to do 1)_ _ _ _ _ _ _ _ _ _ first 4 were W-W-W-L...
18. ### [Combinatorics] # of arithmetic problems possible

[Combinatorics] # of arithmetic problems possible <SOLVED> "How many arithmetic problems of the following form are possible? You must use each of the digits 1 through 9, they must appear in numerical order from left to right, and you can use any combination of the + and * symbols you like, as...
19. ### enumerative combinatorics

Hi, can anybody please help me or analyse this question for me please? I really have no idea how to even approach it. It says: Derive the exponential generating function (egf) for permutations having an even number of cycles and that of permutations having an odd number of cycles. There should...
20. ### Hard combinatorics question

Let's say that I have a chess board (64 squares), and I want to color it using 5 colors (green, red, blue, yellow and black), but making sure that neither of the colors touch each other. How many different boards can I make? To clarify a bit: let's say that the square b2 is blue. Then the...