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Expert: - 6/1/2010


Question
In how many ways can 4 balls be put in 5 boxes if
a)the balls are the same color and the boxes are identical
b)the balls are the same color but the boxes are not identical
c)the balls are of different colors but the boxes are identical
d)the balls are of different colors and the boxes are not identical

Answer
a) The ways to do it are:
[1] all the balls can all be put in one box,
[2] three can be put in one with one in another,
[3] two can be put in one with two in another,
[4] two can be put in one with one each in two others,
[5] or all four can be put in different boxes.
That looks like 5 choices.

From here, C(a,b) means "a choose b".
For example, C(5,2) = "5 choose 2" = 5!/(2!(5-2)!) = 5!/(2!3!).
Now 5!/3! = 5*4 and 2! = 2*1, so this is = 5*4/(2*1) = 20/2 = 10.

b) This makes the above choices have differing numbers of ways of doing them.
The number of ways for [1] 5 different boxes, so choosing 1 box to put them all in is out of 5,
so it is C(5,1)=5; [2] pick 1 of 5 to put the first four in and 1 of 4 for the last, so it is
C(5,1)*C(4,1)=5*4=20; [3] pick one box for the pair out of five and then 2 more out of 4 for the additional balls, givein C(5,1)*C(4,2)=5*6=30;  the final answer would be found by adding up the number of combinations for [1], [2],[3], [4], and [5].

c) If the balls are different with the boxes the same,
[1] for this one, since it maters not which box is chosen, the number would be 1;
[2] the color means how many ways can the color of balls be split, and that is C(4,1) = 4;
[3] again, splitting color is C(4,2) = 6;
[4] there is C(4,2) ways of putting two balls in the first box and it doesn't matter where the others are put, so the number is still 6;
[5] since the boxes are all the same, it matters not which box each ball is placed in,
so the ways of this is 1.
Again, to get the total, add the number from [1], [2], [3], [4], and [5] together.

d) For this one, each ball can go in eac box, so that is a simple 4^5.
Now 4*4 = 16, 4*16 = 64, 4*64 = 256, and 4*256 = 1,024, and that is 5 4's multiplied together.

If you're into computers, which are in binary, that is 2^10, which is called a kilobyte.
It is so close to a thousand that a kilobyte is called 'a thousand bytes'.
Similarly, a megabyte is 'a thousand kilobytes' and a gigabyte is 'a thousand megabtes'.
In actuallity, they are really 1,024 of the next one down.

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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.

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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.

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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.

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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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