Skip main navigation

Let’s count – Sharings into nonempty sets

Let's count - sharings into nonempty sets
Do your best in trying to solve the following problems. It is normal that this step will take some time, we recommend to perseverate and wait before looking at the complete solution of the exercises in the video or in the pdf below.

Before looking at the exercise let us suggest to convert a problem of looking at the (n)-sharings of (I_k) without empty sets (here (k,ninmathbb N_{ge 1})) into a problem of (k)-sequences of (I_n) where all the elements of (I_n) appear. The formula of their number, equal to

[n^k-binom{n}{n-1} (n-1)^k+binom{n}{n-2}(n-2)^k+cdots +(-1)^{n-1} binom{n}1 1^k]

begins with the number (n^k) of (k) sequences of (I_n). Then you subtract the number(displaystylebinom{n}{n-1}(n-1)^k) of sequences that contain only (n-1) elements of (I_n), and then you add the number(displaystylebinom{n}{n-2}(n-2)^k) of sequences that contain only (n-2) elements of (I_n), and so on…

Exercise 3.

Write how many ways there are to subdivide 4 people into 3 (nonempty) distinguishable groups (A, B, C) in two different ways:
a) by means of a known formula;
b) Count first partitions into 3 nonempty sets and then “label’’ the sets.

© CM
This article is from the free online

Combinatorics: Strategies and Methods for Counting

Created by
FutureLearn - Learning For Life

Reach your personal and professional goals

Unlock access to hundreds of expert online courses and degrees from top universities and educators to gain accredited qualifications and professional CV-building certificates.

Join over 18 million learners to launch, switch or build upon your career, all at your own pace, across a wide range of topic areas.

Start Learning now