Permutations, combinations, bijective proofs, generating functions
1
vote
1answer
27 views
Combinatorics riddle: Sorting people in a cinema line.
Say i want to go to the cinema. There are two types of movies.
Action movie.
Drama movie.
Because action is more interesting it costs 50$. And the cost for drama is 10€.
There are 200 people ...
1
vote
1answer
26 views
Computing the probability of rolling a sum of 18 on 4 six-sided dice
The following PDF gives an explanation on page 11. Unfortunately I do not know how to reproduce it here.
http://web.mit.edu/~qchu/Public/TopicsInGF.pdf
In short, I am not sure how the symmetry ...
0
votes
2answers
35 views
Combinatorics Example Problem
Ten weight lifters are competing in a team weightlifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada.
Part 1
If the scoring ...
2
votes
2answers
44 views
Work-at-home days such that the office is always staffed
I am quite rusty on combinatorial formulas, but I think the following practical question has a combinatorial answer. This is not homework.
A company gives it's technicians two work days per week to ...
1
vote
1answer
31 views
Combinatorial Proof for a Recursive Sequence
For $n>3$ let $g_n = g_{n-1} + g_{n-3}$ where the recursion takes the value 1 for n = 0,1,2.
Prove that $g_{2n} = (g_n)^2 + 2g_{n-2}g_{n-1}$ combinatorially, $n > 1$.
For the time being I am ...
1
vote
2answers
28 views
How many ways can you make 100 out of 1, 5, 10, 25, and 100?
Or rather, what is a generalized way to complete this question? Note that only addition would be used, no multiplying numbers by each other (though that would be an interesting problem in itself).
...
1
vote
2answers
36 views
Multinomial Theorem - Combinatorics Question
If 8 new teachers are to be divided among 4 schools, how many divisions are possible?
I understand that in this question you are just solving for the multinomial coefficients of the multinomial ...
4
votes
4answers
60 views
Evaluating Sums Algebraically or Combinatorially
Consider
(1) $$\sum_{k=0}^{n}\binom{n}{k}2^{k-n}$$
(2) $$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$
These sums appear too difficult (in my mind) to evaluate combinatorially. What are some ...
0
votes
1answer
31 views
How many 9 letter strings are there that contain at least 3 distinct vowels?
Question: How many 9 letter strings are there that contain at least 3 distinct vowels?
I am studying and I was wondering if this answer could be an alternative answer to the question above: ...
1
vote
1answer
24 views
8 friends, 7 nights, invite 4 every night, all of the friends must be invited, how many options?
Assume I have 8 friends, I want to invite 4 friends each night for 7 night so everyone will be invited at least once. How many combinations are there to do it?
I think I'm supposed to use the ...
0
votes
0answers
24 views
Probability of Obtaining A Particular Sum from Successive Dice Rolls
Suppose you have a regular die with 6 faces numbered 1 through 6, respectively, and roll the die 4 times. What is the probability that the sum of the 4 rolls is 14?
This problem is equivalent to ...
0
votes
1answer
34 views
How many ways are there to sit $n$ couples on a bench when every couple sits together?
How many ways are there to sit $n$ couples on a bench with $2n$ sits, when every couple sits together?
How many ways are there to sit the couples so that none of the couples will sit together?
0
votes
1answer
24 views
Construct set family whose members intersect at even number of points
F $\subseteq 2^{[n]}$ is a set family. Every member of F has odd size. Every two distinct members of F intersect at even number of points.
1) Show that |F| $\leq$ n
2) Suppose now every member has ...
6
votes
0answers
45 views
The right way to motivate lattice theory in a combinatorics class
I am attending a course on combinatorics. I was asked to present Möbius functions on lattices for this course. I was trying to look for a simple non-trivial problem that illustrates the need for ...
3
votes
1answer
31 views
Evaluating Sums Combinatorially
Consider the following finite sums:
(1) $\sum k(k!)$ for k from 1 to n
(2) $\sum (k-1)(n-k)$ from 1 to n
I am trying to determine how to evaluate these sums combinatorially. It seems the first is ...
