You are here:

Advanced Math/r-permutations and r-combinations of grouped objects

Advertisement


Question
1. How is the number of r-permutations of n objects including n1,n2,n3, identical objects calculated? (r<n)

For example, what is the number of ways in which 3 yellow balls, 2 red balls and 4 green balls can be put in 5 baskets, each in one basket, using five of the balls. (in this example, n=9 , n1=3 , n2=2 , n3=4 and r=5.)

2. How is the number of r-combinations of such objects calculated? (where order does not matter at all, even between different groups)

Thank you very much in advance.

Answer
Think of 1 as arranging things in order with the borders between the baskets as an object.
Now one basket border has to go at either end.  For 5 baskets side by side, there is one border between each of the baskets and one at either end.  Since there are 5 baskets, there are 5-1=4 borders between that can be thought of as being arranged with the balls.

The number of objects to arrange is then 3 yellow + 2 red + 4 green + 4 borders = 13 objects.  Since there is no particular way that order is involved, that is 13!/(3!2!4!4!).

Cancelling one of the 4! terms makes it 13*12*11*10*9*8*7*6*5/(3*2*1*2*1*4*3*2*1).  
The 1st 3*2 on bottom cancels a 6 on top, next are 2 and 4, which cancel with an 8 on top,
the 1's in the denominator disappear, the last two terms in the denominator are 2 and 3.
Note that 12/(2*3) = 2.  We cancelled a 6, an 8, and made the 12 a 2 while cancelling all of the terms in the denominator.  That leaves 13*2*11*10*9*7*5.  Now if you know that 7*11*13 = 1001, that leaves 2*10*9*5*1001.  Since 2*5 = 19, that is 900*1001, which is 900,900.

That is the number of combinations.  A permutation is an order combination.
Thus, we would think of each of the 9 balls as a separate ball.  Then remember that each of the partitions could go anywhere, which makes for 13 items.  Thus, it would be 13!/4! since
13 - 9 = 4.  That works out to  be 259,459,200.  That is 288 times as many for each un-ordered combination.

Advanced Math

All Answers


Answers by Expert:


Ask Experts

Volunteer


Scott A Wilson

Expertise

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 also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. 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.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.

©2016 About.com. All rights reserved.