# Simple combinatorics problem

## Main Question or Discussion Point

So here is the problem:

A lot of 140 semiconductor chips is inspected by choosing a sample of five chips. Assume 10 of the chips do not conform to customer requirements.

...

c) How many samples of five contain at least one non-conforming chip?
Now what seems logical to me is first choose 1 of the 10 non-performing and then choose 4 from the remaining 139 chips.

What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260.

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Suppose we simplify it a bit so we have 2 non-performing, A and B, and 2 performing, C and D, and we are looking at a sample size of 2. Using your method we first pick A from the 2 non-performing, and then 1 of the remaining 3, giving us:
AB
AC
Now we pick B from the two non-performing, and 1 of the the remaining 3, giving us:
BA
BC
BD

Unfortunately we have counted AB twice, first as AB and then as BA. So that's why your way doesn't work.

To solve it correctly you should take the total number of samples, which is C(140,5), and then subtract the ones that don't have any non-performing elements, which is C(130,5).

Homework Helper
"What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260."

The phrase "at least one" does not mean the same thing as "exactly one". The problem involved at least one, you answered as if it were exactly one.

"What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260."

The phrase "at least one" does not mean the same thing as "exactly one". The problem involved at least one, you answered as if it were exactly one.
If you read closer, he actually made a different mistake. He said "choose 4 from the remaining 139 chips" not "choose 4 from the remaining 130 chips" (which would be the "exactly one" option).