Oscar Burton
2022-10-14
Answered

6 points are located on a circle and lines are drawn connecting these points, each pair of points connected by a single line. What can be the maximum number of regions into which the circle is divided?

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elulamami

Answered 2022-10-15
Author has **22** answers

In general the maximum number of regions you can get from 𝑛 points is given by $$(}\genfrac{}{}{0ex}{}{n}{4}{\textstyle )}+{\textstyle (}\genfrac{}{}{0ex}{}{n}{2}{\textstyle )}+1$$

This can be proved using induction (other combinatorial proofs exist too).

This is an oft cited puzzle to show the perils of generalizing based on first few values. We get powers of $2$ till $n=5$, after which we get $31$.

This can be proved using induction (other combinatorial proofs exist too).

This is an oft cited puzzle to show the perils of generalizing based on first few values. We get powers of $2$ till $n=5$, after which we get $31$.

asked 2021-09-08

A restaurant offers a $12 dinner special with seven appetizer options, 12 choices for an entree, and 6 choices for a dessert. How many different meals are available when you select an appetizer, an entree,and a dessert?

asked 2021-09-09

In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed?

asked 2022-01-06

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win 2 for each black ball selected and we lose 2 for each black ball selected and we lose 1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

asked 2022-09-11

Combinatorial proof of a Fibonacci identity: $n{F}_{1}+(n-1){F}_{2}+\cdots +{F}_{n}={F}_{n+4}-n-3.$

Does anyone know a combinatorial proof of the following identity, where ${F}_{n}$ is the $n$th Fibonacci number?

Does anyone know a combinatorial proof of the following identity, where ${F}_{n}$ is the $n$th Fibonacci number?

asked 2022-06-03

How are we able to calculate specific numbers in the Fibonacci Sequence?

I was reading up on the Fibonacci Sequence, $\text{{1,1,2,3,5,8,13,....}}$ when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position.

Is there a way to do this?

I was reading up on the Fibonacci Sequence, $\text{{1,1,2,3,5,8,13,....}}$ when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position.

Is there a way to do this?

asked 2021-11-05

The reaction

$A+B\text{}yields\text{}C+D$

$rate=k\left[A\right]{\left[B\right]}^{2}$

has an intial rate of$0.00220\frac{M}{s}$

what will the initial rate be if$\left[A\right]$ is halved and $\left[B\right]$ is tripled? Answer must be in $\frac{M}{s}$

What will the intial rate be if$\left[A\right]$ is tripled and $\left[B\right]$ is halved? Answer must be in $\frac{M}{s}$

has an intial rate of

what will the initial rate be if

What will the intial rate be if

asked 2022-11-17

Let $k,p$ be positive integers. Is there a closed form for the sums

$\sum _{i=0}^{p}{\textstyle (}\genfrac{}{}{0ex}{}{k}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k+p-i}{p-i}{\textstyle )}\text{, or}$

$\sum _{i=0}^{p}{\textstyle (}\genfrac{}{}{0ex}{}{k-1}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k+p-i}{p-i}{\textstyle )}\text{?}$

$\sum _{i=0}^{p}{\textstyle (}\genfrac{}{}{0ex}{}{k}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k+p-i}{p-i}{\textstyle )}\text{, or}$

$\sum _{i=0}^{p}{\textstyle (}\genfrac{}{}{0ex}{}{k-1}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k+p-i}{p-i}{\textstyle )}\text{?}$