What is the connection between number of permutations and number of subsets?

Question

How many different ways to fill 100 boxes in a line with black or white balls. (One box can only contain one ball at a time.)

My attempt :

Different ways to fill 1 st box = 2 Different ways to fill 2 nd box = 2 Different ways to fill 3 rd box = 2 ' ' ' ' ' Different ways to fill 99th box = 2 Different ways to fill 100th box = 2

SO by method of counting the answer is $2^{100}$.

This is same as number of subsets a set of 100 elements has?

What is the connection this type of questions and $2^n$ has?

A filling is uniquely determined by the choice of the subset of boxes containing black balls. — Jyrki Lahtonen, Sep 07 '19 at 05:19
You mention permutations in the title but I don't see anything about permutations in your actual question. — bof, Sep 07 '19 at 05:31
Possible duplicate of Number of subsets of a set having r elements — Chain Markov, Sep 07 '19 at 12:36

Derek Luna · Accepted Answer · 2020-03-03T17:42:47.923

3

A subset $B$ of a set $A$ can be thought of by the following scheme:

If $x \in A$ then either $x \in B$ or $x\notin B$. So for each element we have these two choices, and we must make such a choice for each element. This is exactly what you did except in this case the choice was black or white.

edited Mar 03 '20 at 17:42

answered Sep 07 '19 at 05:22

Derek Luna

2,732
8
19

Short and satisfying – Kaj Hansen Sep 07 '19 at 05:28

What is the connection between number of permutations and number of subsets?

1 Answers1