Balls in boxes and multisets 2

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From the course by National Research University Higher School of Economics

Introduction to Enumerative Combinatorics

51 ratings

National Research University Higher School of Economics

Introduction to Enumerative Combinatorics

51 ratings

Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers. The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.

From the lesson

Binomial coefficients, continued. Inclusion and exclusion formula.

In the first part of this lecture we will see more applications of binomial coefficients, in particular, their appearance in counting multisets. The second part is devoted to the principle of inclusion and exclusion: a technique which allows us to find the number of elements in the union of several sets, given the cardinalities of all of their intersections. We discuss its applications to various combinatorial problem, including the computation of the number of permutations without fixed points (the derangement problem).

The cab driver problem17:26

Balls in boxes and multisets 110:20

Balls in boxes and multisets 26:04

Integer compositions11:21

Principle of inclusion and exclusion: two examples12:36

Principle of inclusion and exclusion: general statement9:59

The derangement problem19:22

Meet the Instructors

Evgeny Smirnov
Associate Professor
Faculty of Mathematics

[MUSIC]

So we want to find the number of k element model sets of the sets with n elements or

which is the same number of putting k balls inside n boxes.

To find this number, let us reinterpret this problem in a slightly different way.

Suppose we have such a configuration of balls and boxes.

Let us represent it as follows by a sequence of defined

objects while circles will represent balls and

vertical bars, which will separate boxes.

So the space to the left of the first bar is the first box,

then suppose we're going from left to right.

And at first, we're not looking

at the first bar bounding the lead box from the left and

we see one ball inside the first box.

And then we'll put a bar which separates the first and the second boxes and

then we put another, there are no balls inside the second box and

we put a separator between the second and the third box.

So, we'll put a bar here an then there are three balls inside the third box.

And then there is a bar separator between the third and the fourth box.

And finally, there is one ball in the last, the right most box.

So we get a configuration of, k balls,

And instead of n boxes,

we get n minus 1 separator bars.

So this, this picture innately determines the configuration of balls inside boxes.

So, we need to compute the number of such configurations and

this is the standard problem involving binomial coefficients.

Namely, we have k plus n-1 objects called balls or bars.

And out of these, we need to select k

which corresponds to balls.

Or equivalently, we need to select n-1 corresponding to bars.

So, configurations,

Of balls in boxes.

Correspond to

configuration of,

Bars and balls.

And there are k+n-1 objects.

And out of these,

k are balls and n-1 are bars.

So, the number of such configurations is,

k+n-1 choose k, out of k+n-1 objects.

We need to pick those which correspond to balls.

We get a theorem.

Theorem, k balls can be put,

Into n boxes,

In n+k-1,

choose k, ways.

Equivalently,

The number of k-element

multisets.

In an n-element set

is n+k-1, choose k.

And right now, we will discuss another version of this problem

about compositions of integer number into positive integer assignments.

[MUSIC]

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