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QUESTION: how many word can be arranged from "business" with only 2s are adjacent?
ANSWER: There are 3 letters s. There are 5 other letters, and the way to arrange them is A=5!.
Three letters must be inserted, but two of them must be inserted next to each other. There are six spots we could put them in (between the letters, before, or after), so the number of ways of doing this is C(6,2) = 6!/((6-2)!2!)=B.
Multiply A by B to get the answer.
---------- FOLLOW-UP ----------
QUESTION: Thanks for answering but the answer is 3600. So, what do i need to do?
ANSWER: There are 8 letters, so the ways of arranging them is 8!.
This must be divided by 3! since there are 3 s's.
This means there are 6,720 ways of arranging them.
The number of times where only 2 s's are together, though may take a little while to do.
If 2 of the s's were at the start, there would be 720 ways of arranging the others letters. However, 120 of these ways would put an s next, which would put 3 s's together, so we have 600.
The same is true if 2 s's were at the end, meaning that there would be another 600.
If 2 s's are in the middle, there are two locations the other s can't be in, which means for each spot in the middle there are 480 ways of arranging the other letters.
There are 5 ways of putting them in the middle, and 480 for each time, meaning there are 2,400 ways of putting them in the middle.
Now if you take the number of ways in the middle plus the number of ways of starting plus the number of ways of ending, you get
600 + 2,400 + 600 = 3,600.
I enjoyed that challenge. I really had to think about that one and I loved it. Thanks for correcting me and making me think more in depth about it.
---------- FOLLOW-UP ----------
QUESTION: Thanks but can you explain how do you get the amount of the ways?
There are 8 letters and we have taken out 2 of them (s and s).
That leaves six letters. The number of ways of arranging six letters is 6!=6*5*4*3*2*1=720.
Note that is the two s's are at the start, 1/6 of these ways would put an s as the next letter, so there are only 600.
If the 2 s's were in the middle, we have to account of the other s being at the start of end, which means 1/6 + 1/6 = 2/6 = 1/3.
This leaves only 2/3 of 720, which is 480. There are 5 ways they could go in the middle, so 5*480 is 2400.
I added 600 for the front and 600 for the end to get
600 + 2,400 + 600 = 3,600.
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Answers by Expert:
Scott A Wilson
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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?
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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later.
I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman.
I tutored at Mathnasium for well over a year.
I worked at The Boeing Company for over 5 years.
I received an MS degreee in Mathematics from Oregon State Univeristy.
The classes I took were over 100 hours of upper division credits in mathematical courses such as
calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more.
I graduated with honors in both my BS and MS degrees.
Past/Present Clients: College Students at Oregon State University, various math people since college,
over 7,500 people on the PC from the US and rest the world.
Publications
My master's paper was published in the OSU journal.
The subject of it was Numerical Analysis used in shock waves and rarefaction fans.
It dealt with discontinuities that arose over time.
They were solved using the Leap Frog method.
That method was used and improvements of it were shown.
The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.
Education/Credentials
Master of Science at OSU with high honors in mathematics.
Bachelor of Science at OSU with high honors in mathematical sciences.
This degree involved mathematics, statistics, and computer science.
I also took sophmore level physics and chemistry while I was attending college.
On the side I took raquetball, but that's still not relevant.
Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State.
I was in near the top in most of my classes. In several classes in mathematics, I was first.
In a class of over 100 students, I was always one of the first ones to complete the test.
I graduated with well over 50 credits in upper division mathematics.
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My clients have been students at OSU, people nearby, friends with math questions,
and several people every day on the PC, and you're probably make one more.
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