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Hi all, I have an issue understanding a statement I read in my text.

It first states the following Proposition (Let's call it Proposition A):

The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n \choose r##. In particular,

$$2^n = \sum_{k = 0}^{n} {n \choose k}$$

is the number of subsets of a set of ##n## objects.

##\textbf{Question 1}##: What does "number of subsets of a set of ##n## objects" mean? Does it mean the number of ways I can split n objects into "2 groups/subsets"?

It later (in a separate section) states another Proposition (B):

The number of ways that ##n## objects can be grouped into ##r## classes with ##n_i## in the ##i##th class, ##i = 1,..,n##, and ##\sum_{i=1}^{r} n_i = n## is

$${n \choose n_1 n_2 ... n_r} = \frac{n!}{n_1 ! n_2 ! ...n_r !}$$

Proposition A is the special case for ##r = 2##

##\textbf{Question 2}##: What does "Proposition A is the special case for ##r = 2##" mean?

Many thanks in advance.

Hi all, I have an issue understanding a statement I read in my text.

It first states the following Proposition (Let's call it Proposition A):

The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n \choose r##. In particular,

$$2^n = \sum_{k = 0}^{n} {n \choose k}$$

is the number of subsets of a set of ##n## objects.

##\textbf{Question 1}##: What does "number of subsets of a set of ##n## objects" mean? Does it mean the number of ways I can split n objects into "2 groups/subsets"?

It later (in a separate section) states another Proposition (B):

The number of ways that ##n## objects can be grouped into ##r## classes with ##n_i## in the ##i##th class, ##i = 1,..,n##, and ##\sum_{i=1}^{r} n_i = n## is

$${n \choose n_1 n_2 ... n_r} = \frac{n!}{n_1 ! n_2 ! ...n_r !}$$

Proposition A is the special case for ##r = 2##

##\textbf{Question 2}##: What does "Proposition A is the special case for ##r = 2##" mean?

Many thanks in advance.

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