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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
?
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
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan Reply
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
Chine Reply
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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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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