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Just carpet has eight different quality grades of carpeting. Three types are being selected for a corporate test on durability. how many different samples of size 3 can be taken without replacement? how many different samples of size 5 can be taken without replacement?

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Answer:

When we have a set of N elements, the number of combinations of K elements (such that N ≤ K) from these N elements is:

[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]

Then if we have 8 different types of carpet, and we want to see how many different samples of 3 types we can choose, we just need to replace N by 8, and K by 3 in the above equation:

[tex]C(8, 3) = \frac{8!}{(8 - 3)!*3!} = \frac{8!}{5!*3!} = \frac{8*7*6}{3*2} = 56[/tex]

So there are 56 different samples.

Now we can do the same, but this time we want to use 5 types of carpet, then we will have K = 5.

[tex]C(8, 5) = \frac{8!}{(8 - 5)!*5!} = \frac{8!}{3!*5!} = 56[/tex]

Again, we have 56 different samples.

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