Question

a set of distinct elements {x1, x2, x3.... , xn} . and you draw at random with replacement n elements samples, how many distinct elements samples can be created? example suppose you have {a,b,c} then sample with replacement = {a,a, a} , {a,a,b,}, {b,b,b}

Answer 1

Number of distinct elements in the set = n : (x1,x2,x3,...,xn)

you draw at random with replacement n elements

1st element in the sample can be any of the n elements (x1,x2,x3,..,xn):

2nd element in the sample can also be any of the n elements (x1,x2,x3,..,xn):

....

...

nth element in the sample can also be any of the n elements (x1,x2,x3,..,xn):

Number of distinct element samples that can be created = n x n x......n (n times) = nⁿ

Number of distinct element samples that can be created = nⁿ

For

three element {a,b,c} ; All possible 3³ =27 distinct element samples are as follows

1	a	a	a
2	a	a	b
3	a	a	c
4	a	b	a
5	a	b	b
6	a	b	c
7	a	c	a
8	a	c	b
9	a	c	c
10	b	a	a
11	b	a	b
12	b	a	c
13	b	b	a
14	b	b	b
15	b	b	c
16	b	c	a
17	b	c	b
18	b	c	c
19	c	a	a
20	c	a	b
21	c	a	c
22	c	b	a
23	c	b	b
24	c	b	c
25	c	c	a
26	c	c	b
27	c	c	c