All Questions
20
questions
4
votes
3
answers
272
views
How many ways for a beetle to move from bottom left corner to upper right corner in a 6 x 6 grid if it must be done in 14 steps only?
Note: We allow all four directions (up, down, right, left but no diagonal)
The 6 x 6 grid is composed of 7 horizontal lines and 7 vertical lines. We are to calculate how many 14 steps paths are ...
0
votes
1
answer
156
views
How is the diagonal constraint in lattice path needed for the Catalan proofs?
I have been reading about the Catalan numbers and how they are they appear in many problems such as:
lattice paths
valid pair of parenthesis
mountains with up/downstrokes
non-crossing handshakes
...
- 1,609
2
votes
1
answer
136
views
Why are the catalan numbers giving the unique/correct patterns from all the combinations?
I am reading about catalan numbers and they are considered to represent the number of valid pair of parentesis, mountains etc.
Although the number checks out correct when comparing against specific ...
- 1,609
6
votes
0
answers
111
views
To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $p:q:r = 1:7:35$.
To find $n$ such that the expansion of $(1+x)^n$ has three consecutive coefficients $p,q,r$ that satisfy $$p:q:r = 1:7:35$$
My work:
Suppose the consecutive coefficients are $\binom{n}{k-1}, \binom{n}...
- 12.7k
4
votes
3
answers
198
views
What is the probability of picking a full set from multiset after $m$ draws?
Suppose a bag contains $n$ balls labeled from $1$ to $n$, and suppose I have $k$ of these bags. If I open all of these $k$ bags into an urn, then the urn is effectively a multiset with $kn$ elements: $...
- 7,283
1
vote
3
answers
71
views
A question involving the expansion of $(x+2)^n$
I have the following question:
In the expansion of $(x+2)^n$, $n$ is a positive integer greater than $2$. Given that the ratio between the coefficient of the $x^3$ term and the coefficient of the $x$ ...
- 2,846
16
votes
5
answers
357
views
Intuition behind sums of sums of whole numbers
So I was playing around, and all this is just a curiosity and nothing serious.
Anyway, most readers probably know:
$$1+2+3+4+5+...+(n-1)+n=\frac{1}{2}n^{2}+\frac{1}{2}n=\binom{n+1}{n-1}$$
I started ...
- 672
1
vote
1
answer
39
views
Why does $\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$ computes the probability of $n$ head or tails
$\sum_{i=n}^{2n-1}\binom{i-1}{n-1}2^{1-i}$
For $i = n,n+1,\ldots, 2n - 1$, the sum above computes $P(E_i)$, the probability that i
tosses of a fair coin are required before obtaining $n$ heads or $n$ ...
user675768
0
votes
3
answers
110
views
I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$
I have this identity that I'd like to prove.
$$\displaystyle{\sum_{k=0}^{n}\bigg(\dfrac{n-2k}{n}\binom{n}{k}}\bigg)^2=\dfrac{2}{n}\binom{2n-2}{n-1}$$
Here's what I have done so far: (using a binomial ...
user675768
0
votes
2
answers
88
views
Show that $\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$
Show that if $n\equiv 0\pmod 6$ (although the statement holds true for $n\equiv 0\pmod 3$)
$\binom{n}{1}-3\binom{n}{3}+3^2\binom{n}{5}\cdots=0$
I am having trouble finding the appropriate polynomial ...
user675768
23
votes
5
answers
1k
views
New Year Maths $2019$
$$\;\;\;\color{red}{\binom {20}{19}}\\
\color{orange}{+\binom {19}{a}
+{\binom ab}}\\
\color{green}{+\binom bc
+\binom cd}\\
\color{blue}{+\binom de+
\binom ef}\\
\color{purple}{+\binom fg+\binom gh}\...
2
votes
0
answers
108
views
Simplify a weighted average with enumerations as weights
Could you help me simplify the following expression:
$$\forall n_0 \ge 0, \forall m_1,m_2 \gt 0, \forall k_1 \in [[0,n_0m_1]],\forall k_2 \in [[0,n_0m_2]],$$
$$\overline{k_0}_{(n_0,m_1,m_2)}(k_1,k_2) =...
- 203
2
votes
1
answer
3k
views
Probability of winning a prize in a raffle (each person can only win once)
So I'm trying to figure out the odds of winning a prize in a raffle where each person can buy multiple tickets but you can only win once.
My problem is the following:
There are 2600 tickets sold
...
0
votes
5
answers
2k
views
If I have 10 different pairs of socks and have washed 10 socks, what are the chances that none will match?
I have 10 pairs of different types of socks. I randomly (let's just assume it was true randomness) washed 10 individual socks. It turns out none of them match! What are the chances of this?
I've ...
- 133
0
votes
2
answers
1k
views
How many routes are there that pass through at most one congested intersection
I am trying to solve the following problem, but i am not quite sure how to attack.
Problem Description
A taxi drives from the intersection labeled A to the intersection labeled B in the grid of ...
- 5,768