Question

How toDraw all the possible sample of size 4 of seven random numbers between 0-9 with without replacement method.

Answer 1

Numbers between 0-9 is 1,2,3,4,5,6,7,8

Total 8 numbers

So, N = 8

and we have to form a sample of size 4

So, total number of samples that can be form without replacement = $^NC_n=^8C_4=70$

Samples:

$1234, 1235, 1236, 1237, 1238, 1245, 1246, 1247, 1248, 1256,\\ 1257, 1258, 1267, 1268, 1278, 1345, 1346, 1347, 1348, 1356,\\ 1357, 1358, 1367, 1368, 1378, 1456, 1457, 1458, 1467, 1 4 6 8, \\1 4 7 8, 1 5 6 7, 1 5 6 8, 1 5 7 8, 1 6 7 8, 2 3 4 5, 2 3 4 6, 2 3 4 7, 2 3 4 8, 2 3 5 6, \\2 3 5 7, 2 3 5 8, 2 3 6 7, 2 3 6 8, 2 3 7 8, 2 4 5 6, 2 4 5 7, 2 4 5 8, 2 4 6 7, 2 4 6 8, \\2 4 7 8, 2 5 6 7, 2 5 6 8, 2 5 7 8, 2 6 7 8, 3 4 5 6, 3 4 5 7, 3 4 5 8, 3 4 6 7, 3 4 6 8,\\ 3 4 7 8, 3 5 6 7, 3 5 6 8, 3 5 7 8, 3 6 7 8, 4 5 6 7, 4 5 6 8, 4 5 7 8, 4 6 7 8, 5 6 7 8$

Total 70 samples of size 4 (without replacement)

And, total number of samples that can be form with replacement = $\bar C (n,r)=\dfrac{(r+n-1)!}{r!\times (n-1)!}$

So, $\bar C(8,4)=\dfrac{39916800}{24\times 5040}=330$

Samples:

$1 1 1 1, 1 1 1 2, 1 1 1 3, 1 1 1 4, 1 1 1 5, 1 1 1 6, 1 1 1 7, 1 1 1 8, 1 1 2 2, 1 1 2 3, 1 1 2 4, 1 1 2 5, 1 1 2 6, 1 1 2 7, \\1 1 2 8, 1 1 3 3, 1 1 3 4, 1 1 3 5, 1 1 3 6, 1 1 3 7, 1 1 3 8, 1 1 4 4, 1 1 4 5, 1 1 4 6, 1 1 4 7, 1 1 4 8, 1 1 5 5, 1 1 5 6,\\ 1 1 5 7, 1 1 5 8, 1 1 6 6, 1 1 6 7, 1 1 6 8, 1 1 7 7, 1 1 7 8, 1 1 8 8, 1 2 2 2, 1 2 2 3, 1 2 2 4, 1 2 2 5, 1 2 2 6, 1 2 2 7,\\ 1 2 2 8, 1 2 3 3, 1 2 3 4, 1 2 3 5, 1 2 3 6, 1 2 3 7, 1 2 3 8, 1 2 4 4, 1 2 4 5, 1 2 4 6, 1 2 4 7, 1 2 4 8, 1 2 5 5, 1 2 5 6,\\ 1 2 5 7, 1 2 5 8, 1 2 6 6, 1 2 6 7, 1 2 6 8, 1 2 7 7, 1 2 7 8, 1 2 8 8, 1 3 3 3, 1 3 3 4, 1 3 3 5, 1 3 3 6, 1 3 3 7, 1 3 3 8,\\ 1 3 4 4, 1 3 4 5, 1 3 4 6, 1 3 4 7, 1 3 4 8, 1 3 5 5, 1 3 5 6, 1 3 5 7, 1 3 5 8, 1 3 6 6, 1 3 6 7, 1 3 6 8, 1 3 7 7, 1 3 7 8,\\ 1 3 8 8, 1 4 4 4, 1 4 4 5, 1 4 4 6, 1 4 4 7, 1 4 4 8, 1 4 5 5, 1 4 5 6, 1 4 5 7, 1 4 5 8, 1 4 6 6, 1 4 6 7, 1 4 6 8, 1 4 7 7,\\ 1 4 7 8, 1 4 8 8, 1 5 5 5, 1 5 5 6, 1 5 5 7, 1 5 5 8, 1 5 6 6, 1 5 6 7, 1 5 6 8, 1 5 7 7, 1 5 7 8, 1 5 8 8, 1 6 6 6, 1 6 6 7,\\ 1 6 6 8, 1 6 7 7, 1 6 7 8, 1 6 8 8, 1 7 7 7, 1 7 7 8, 1 7 8 8, 1 8 8 8, 2 2 2 2, 2 2 2 3, 2 2 2 4, 2 2 2 5, 2 2 2 6, 2 2 2 7,\\ 2 2 2 8, 2 2 3 3, 2 2 3 4, 2 2 3 5, 2 2 3 6, 2 2 3 7, 2 2 3 8, 2 2 4 4, 2 2 4 5, 2 2 4 6, 2 2 4 7, 2 2 4 8, 2 2 5 5, 2 2 5 6,\\ 2 2 5 7, 2 2 5 8, 2 2 6 6, 2 2 6 7, 2 2 6 8, 2 2 7 7, 2 2 7 8, 2 2 8 8, 2 3 3 3, 2 3 3 4, 2 3 3 5, 2 3 3 6, 2 3 3 7, 2 3 3 8,\\ 2 3 4 4, 2 3 4 5, 2 3 4 6, 2 3 4 7, 2 3 4 8, 2 3 5 5, 2 3 5 6, 2 3 5 7, 2 3 5 8, 2 3 6 6, 2 3 6 7, 2 3 6 8, 2 3 7 7, 2 3 7 8,\\ 2 3 8 8, 2 4 4 4, 2 4 4 5, 2 4 4 6, 2 4 4 7, 2 4 4 8, 2 4 5 5, 2 4 5 6, 2 4 5 7, 2 4 5 8, 2 4 6 6, 2 4 6 7, 2 4 6 8, 2 4 7 7,\\ 2 4 7 8, 2 4 8 8, 2 5 5 5, 2 5 5 6, 2 5 5 7, 2 5 5 8, 2 5 6 6, 2 5 6 7, 2 5 6 8, 2 5 7 7, 2 5 7 8, 2 5 8 8, 2 6 6 6, 2 6 6 7,\\ 2 6 6 8, 2 6 7 7, 2 6 7 8, 2 6 8 8, 2 7 7 7, 2 7 7 8, 2 7 8 8, 2 8 8 8, 3 3 3 3, 3 3 3 4, 3 3 3 5, 3 3 3 6, 3 3 3 7, 3 3 3 8,\\ 3 3 4 4, 3 3 4 5, 3 3 4 6, 3 3 4 7, 3 3 4 8, 3 3 5 5, 3 3 5 6, 3 3 5 7, 3 3 5 8, 3 3 6 6, 3 3 6 7, 3 3 6 8, 3 3 7 7, 3 3 7 8,\\ 3 3 8 8, 3 4 4 4, 3 4 4 5, 3 4 4 6, 3 4 4 7, 3 4 4 8, 3 4 5 5, 3 4 5 6, 3 4 5 7, 3 4 5 8, 3 4 6 6, 3 4 6 7, 3 4 6 8, 3 4 7 7,\\ 3 4 7 8, 3 4 8 8, 3 5 5 5, 3 5 5 6, 3 5 5 7, 3 5 5 8, 3 5 6 6, 3 5 6 7, 3 5 6 8, 3 5 7 7, 3 5 7 8, 3 5 8 8, 3 6 6 6, 3 6 6 7,\\ 3 6 6 8, 3 6 7 7, 3 6 7 8, 3 6 8 8, 3 7 7 7, 3 7 7 8, 3 7 8 8, 3 8 8 8, 4 4 4 4, 4 4 4 5, 4 4 4 6, 4 4 4 7, 4 4 4 8, 4 4 5 5,\\ 4 4 5 6, 4 4 5 7, 4 4 5 8, 4 4 6 6, 4 4 6 7, 4 4 6 8, 4 4 7 7, 4 4 7 8, 4 4 8 8, 4 5 5 5, 4 5 5 6, 4 5 5 7, 4 5 5 8, 4 5 6 6,\\ 4 5 6 7, 4 5 6 8, 4 5 7 7, 4 5 7 8, 4 5 8 8, 4 6 6 6, 4 6 6 7, 4 6 6 8, 4 6 7 7, 4 6 7 8, 4 6 8 8, 4 7 7 7, 4 7 7 8, 4 7 8 8,\\ 4 8 8 8, 5 5 5 5, 5 5 5 6, 5 5 5 7, 5 5 5 8, 5 5 6 6, 5 5 6 7, 5 5 6 8, 5 5 7 7, 5 5 7 8, 5 5 8 8, 5 6 6 6, 5 6 6 7, 5 6 6 8,\\ 5 6 7 7, 5 6 7 8, 5 6 8 8, 5 7 7 7, 5 7 7 8, 5 7 8 8, 5 8 8 8, 6 6 6 6, 6 6 6 7, 6 6 6 8, 6 6 7 7, 6 6 7 8, 6 6 8 8, 6 7 7 7,\\ 6 7 7 8, 6 7 8 8, 6 8 8 8, 7 7 7 7, 7 7 7 8, 7 7 8 8, 7 8 8 8, 8 8 8 8$

Total 330 Samples of size 4 (with replacement)