Definition:

The sequence of random variables is said to be exchangeable if the distribution of the vector is the same as that of , for any permutation of elements.

Suppose we condition on the sample of distinct observations , there are as many different samples as there are ways of choosing objects out of a set of possible contenders, repetitions being allowed.

At this point it is interesting to introduce a new
notation for a bootstrap resample,
up to now we have noted a possible
reasample, say
,
because of the exchangeability/symmetry property
we can recode this as the vector counting
the number of occurrences of each of the observations.
in this recoding we have
and the set of all bootstrap resamples
is the dimensional simplex

Here is the argument I used in class to explain how big is. Each component in the vector is considered to be a box, there are boxes to contain balls in all, we want to contain to count the number of ways of separating the n balls into the boxes. Put down separators of to make boxes, and balls, there will be positions from which to choose the bars' positions, for instance our vector above corresponds to:

Stirling's formula ( ) gives an approximation ,

here is the function file `approxcom.m`

function out=approxcom(n) out=round((pi*n)^(-.5)*2^(2*n-1));that produces the following table of the number of resamples:

Are all these samples equally likely, thinking about the probability of drawing the sample of all 's by choosing the index times in the integer uniform generation should persuade you that this sample appears only once in times. Whereas the sample with once and all the other observations can appear in out of the ways.