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What are the generalisations of the pigeon hole principle?
Hahaeileen
this is the person who asked the question.
there are 3 questions:
a) Mr Wong bought some presents for his pupils in his class. He bought some pens, erasers and files. Every pupil is allowed to choose 2 presents. What is the least number of pupils to ensure that two of them will choose the same type of presents?
Assume that the first pupil takes a pen and an eraser, the second pupil takes a pen and a file, and the third pupil takes a file and an eraser, than all the ways of choosing the presents will be up, so when added one more pupil which is 4 pupils altogether, we can ensure that two of them will choose the same type of presents.
b) A box contains 10 blue ice-cream sticks, 20 red ice-cream sticks, 8 yellow ice-cream sticks, 15 green ice-cream sticks and 25 purple ice-cream sticks. How many must I choose to ensure that 12 ice-cream sticks are of the same colour?
When we choose all 10 blue ice-cream sticks, 11 red, all 8 yellow, 11 green and 11 purple, we do not have 12 of anything yet, but if we choose any one remaining, we will have 12 of that colour. Thus, 10+11+8+11+11+1=52 will guarantee at least 12 ice-cream sticks are of the same colour.
c) Show that if we select 51 distinct numbers from 1 to 100, at least two are consecutive numbers. (Hint: You can prove by contradiction)
Because in 100 numbers we can select 50 all odd or 50 all even and there will be no consecutive pair, picking another number must be of the opposite parity and so it will be consecutive with one of your first 50 numbers chosen.
and i have to state the generalistions from the solutions above. please help!
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What are the learning points of the Pigeonhole principle?
The main idea is the most important one! If you have fewer pigeon holes than pigeons and you put every pigeon in a pigeon hole, then there must result at least one pigeon hole with more than one pigeon. Expand the idea perhaps using a finite example. 21 pigeons and 10 holes. Try and see how at least one hole must have more than 2 pigeons. Then think about generalizing it. The real power comes from seeing this all in action. A commonly used example is: Among any N positive integers, there exists 2 whose difference is divisible by N-1.
What is a pigeonhole principle?
The simple form of it states: If m pigeons are put into m pigeonholes, there is an empty hole iff there's a hole with more than one pigeon. In more formal math language it says: Let |A| denote the number of elements in a finite set A ( also known as its cardinality). For two finite sets A and B, there exists a 1-1 correspondence f: A -->B if and only if |A| = |B|.
Use pigeonhole principle to show that one of any n consecutive integers divisible by n?
Let's look at a simple example first such as 3 consecutive integers. We want to show that it is divisible by 3.Take 4,5, and 6 and of course 6 is divisible by 3.The reason for this is can be seen using the pigeonhole principle.When an integer is divided by 3, possible remainders are 0, 1, and 2. It follows that everyinteger can be expressed in one of the forms 3k, 3k + 1, and 3k + 2 where k is an integer.So if we have any three consecutive integers, one of them must be divisible by 3.Let's look at how the pigeonhole principle applies. Suppose we have 3 consecutive integers are non are divisible by 3. Think of the pigeon holes as 3k, 3k + 1, and 3k + 2, now this means no numbers are in the 3k hole and two of them must be in either the 3k+1 hole or the 3k+2 hole. But this contradicts that they are consecutive integers.So for any n, let the pigeon holes be nk, nk+1,... nk+(n-1) and these exhaust the multiples of n. Now if you take n consecutive numbers, you must have a least 1 number in the nk pigeon hole or else they are not consecutive.
What is a mathematical principle?
Counting Principle is one of them
What mathematical principle is equal to 3.14?
That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.
What is the home of pigeon?
A pigeon hole :)
What are the learning points of the Pigeonhole principle?
The main idea is the most important one! If you have fewer pigeon holes than pigeons and you put every pigeon in a pigeon hole, then there must result at least one pigeon hole with more than one pigeon. Expand the idea perhaps using a finite example. 21 pigeons and 10 holes. Try and see how at least one hole must have more than 2 pigeons. Then think about generalizing it. The real power comes from seeing this all in action. A commonly used example is: Among any N positive integers, there exists 2 whose difference is divisible by N-1.
What is a ten letter hyphenated words?
Pigeon-Hole. JT
What is pigeon heart condition?
a hole between the right and left ventricle
What are the names of the members of Canadian band Pigeon Hole?
The Canadian band Pigeon Hole consisted of Dusty Melo and Marmalade. The band was started in 2010 in Canada. It is considered a Canadian Hip Hop band.
Where can one find instructions on how to build a pigeon hole?
Garden Web is a great resource for people to find instructions on how to build a pigeon hole. They have several options and provide printable versions for people to have handy.
What is pigeon hole problems?
Learn to "pigeon hole" problems. Pigeon holes are small compartments in which pet pigeons can make nests. When you pigeon hole your problems, you keep them in the situation in which they occurred. For example, suppose you had a dreadful day at work. The machine you ran broke down and you didn't accomplish your goals for the day. You should do your best to keep this problem at work so it doesn't interfere with your friendships, home life, studies, and other activities.
What is the pigeon hole theory of tort?
The "pigeon hole theory" w.r.t torts refers to Salmonds theory that any 'harm' in order to constitute legal injury must fit into pre-determined 'pigeon holes'. This is in direct contrast with Winfield's theory of all civil harms being actionable.
What is Ricardo's theory of rent?
well a$$ hole. a pigeon just ran into the window
What are the release dates for O-S-S- - 1957 Operation Pigeon Hole 1-13?
O-S-S- - 1957 Operation Pigeon Hole 1-13 was released on: USA: 19 December 1957
What actors and actresses appeared in Horse in the Hole - 2013?
The cast of Horse in the Hole - 2013 includes: Damien Devaney as Charlie Pigeon Kian Murphy as Boy
What does pigeon Hole'' mean?
To put an object in a safe location where it can be found easily.
