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If a coin is tossed six times, in how many ways can it fall four heads and two tails?

First we solve this problem using [link] technique–permutations with similar elements.

We need 4 heads and 2 tails, that is

HHHHTT size 12{ ital "HHHHTT"} {}

There are 6 ! 4 ! 2 ! = 15 size 12{ { {6!} over {4!2!} } ="15"} {} permutations.

Now we solve this problem using combinations.

Suppose we have six spots to put the coins on. If we choose any four spots for heads, the other two will automatically be tails. So the problem is simply

6C4 = 15 size 12{6C4="15"} {} .

Incidentally, we could have easily chosen the two tails, instead. In that case, we would have gotten

6C2 = 15 size 12{6C2="15"} {} .

Further observe that by definition

6C4 = 6 ! 2 ! 4 ! size 12{6C4= { {6!} over {2!4!} } } {}

and 6C2 = 6 ! 4 ! 2 ! size 12{6C2= { {6!} over {4!2!} } } {}

Which implies

6C4 = 6C2 size 12{6C4=6C2} {} .
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Combinations: involving several sets

So far we have solved the basic combination problem of r size 12{r} {} objects chosen from n size 12{n} {} different objects. Now we will consider certain variations of this problem.

How many five-people committees consisting of 2 men and 3 women can be chosen from a total of 4 men and 4 women?

We list 4 men and 4 women as follows:

M 1 M 2 M 3 M 4 W 1 W 2 W 3 W 4 size 12{M rSub { size 8{1} } M rSub { size 8{2} } M rSub { size 8{3} } M rSub { size 8{4} } W rSub { size 8{1} } W rSub { size 8{2} } W rSub { size 8{3} } W rSub { size 8{4} } } {}

Since we want 5-people committees consisting of 2 men and 3 women, we'll first form all possible two-man committees and all possible three-woman committees. Clearly there are 4C2 = 6 two-man committees, and 4C3 = 4 three-woman committees, we list them as follows:

2-Man Committees 3-Woman Committees
M 1 M 2 size 12{M rSub { size 8{1} } M rSub { size 8{2} } } {} W 1 W 2 W 3 size 12{W rSub { size 8{1} } W rSub { size 8{2} } W rSub { size 8{3} } } {}
M 1 M 3 size 12{M rSub { size 8{1} } M rSub { size 8{3} } } {} W 1 W 2 W 4 size 12{W rSub { size 8{1} } W rSub { size 8{2} } W rSub { size 8{4} } } {}
M 1 M 4 size 12{M rSub { size 8{1} } M rSub { size 8{4} } } {} W 1 W 3 W 4 size 12{W rSub { size 8{1} } W rSub { size 8{3} } W rSub { size 8{4} } } {}
M 2 M 3 size 12{M rSub { size 8{2} } M rSub { size 8{3} } } {} W 2 W 3 W 4 size 12{W rSub { size 8{2} } W rSub { size 8{3} } W rSub { size 8{4} } } {}
M 2 M 4 size 12{M rSub { size 8{2} } M rSub { size 8{4} } } {}
M 3 M 4 size 12{M rSub { size 8{3} } M rSub { size 8{4} } } {}

For every 2-man committee there are four 3-woman committees that can be chosen to make a 5-person committee. If we choose M 1 M 2 as our 2-man committee, then we can choose any of W 1 W 2 W 3 , W 1 W 2 W 4 , W 1 W 3 W 4 , or W 2 W 3 W 4 as our 3-woman committees. As a result, we get

M 1 M 2 , W 1 W 2 W 3 M 1 M 2 , W 1 W 2 W 4 M 1 M 2 , W 1 W 3 W 4 M 1 M 2 , W 2 W 3 W 4

Similarly, if we choose M 1 M 3 as our 2-man committee, then, again, we can choose any of W 1 W 2 W 3 , W 1 W 2 W 4 , W 1 W 3 W 4 , or W 2 W 3 W 4 as our 3-woman committees.

M 1 M 3 , W 1 W 2 W 3 M 1 M 3 , W 1 W 2 W 4 M 1 M 3 , W 1 W 3 W 4 M 1 M 3 , W 2 W 3 W 4

And so on.

Since there are six 2-man committees, and for every 2-man committee there are four 3- woman committees, there are altogether 6 4 = 24 five-people committees.

In essence, we are applying the multiplication axiom to the different combinations.

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A high school club consists of 4 freshmen, 5 sophomores, 5 juniors, and 6 seniors. How many ways can a committee of 4 people be chosen that includes

  1. One student from each class?
  2. All juniors?
  3. Two freshmen and 2 seniors?
  4. No freshmen?
  5. At least three seniors?
  1. Applying the multiplication axiom to the combinations involved, we get

    4C1 5C1 5C1 6C1 = 600 size 12{4C1 cdot 5C1 cdot 5C1 cdot 6C1="600"} {}
  2. We are choosing all 4 members from the 5 juniors, and none from the others.

    5C4 = 5 size 12{5C4=5} {}
  3. 4C2 6C2 = 90 size 12{4C2 cdot 6C2="90"} {}

  4. Since we don't want any freshmen on the committee, we need to choose all members from the remaining 16. That is

    16 C4 = 1820 size 12{"16"C4="1820"} {}
  5. Of the 4 people on the committee, we want at least three seniors. This can be done in two ways. We could have three seniors, and one non-senior, or all four seniors.

    6C3 14 C1 + 6C4 = 295 size 12{6C3 cdot "14"C1+6C4="295"} {}
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How many five-letter word sequences consisting of 2 vowels and 3 consonants can be formed from the letters of the word INTRODUCE?

First we select a group of five letters consisting of 2 vowels and 3 consonants. Since there are 4 vowels and 5 consonants, we have

4C2 5C3 size 12{4C2 cdot 5C3} {}

Since our next task is to make word sequences out of these letters, we multiply these by 5 ! size 12{5!} {} .

4C2 5C3 5 ! = 7200 size 12{4C2 cdot 5C3 cdot 5!="7200"} {} .

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A standard deck of playing cards has 52 cards consisting of 4 suits each with 13 cards. In how many different ways can a 5-card hand consisting of four cards of one suit and one of another be drawn?

We will do the problem using the following steps. Step 1. Select a suit. Step 2. Select four cards from this suit. Step 3. Select another suit. Step 4. Select a card from that suit.

Applying the multiplication axiom, we have

Ways of selecting a suit Ways if selecting 4 cards from this suit Ways if selecting the next suit Ways of selecting a card from that suit
4C1 size 12{4C1} {} 13 C4 size 12{"13"C4} {} 3C1 size 12{3C1} {} 13 C1 size 12{"13"C1} {}
4 C 1 13 C 4 3 C 1 13 C 1 = 111,540 .
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Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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what is a peer
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What is meant by 'nano scale'?
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What is STMs full form?
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scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
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there is no specific books for beginners but there is book called principle of nanotechnology
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how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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