In general if you have $n$ objects, and you want to choose $k$ of them, there are $\binom{n}{k}$ ways to do so, where $$ \binom{n}{k} = \frac{n!}{k! (n-k)!}.$$
So in your case $n= 140$ is the number of chips, and we want to take samples of size $k=5$, so there are $\binom{140}{5} = 416,965,528$ ways to sample the chips.
To count the number of samples (of size 5) containing exactly one of the 10 faulty chips: first select the faulty chip, there are $\binom{10}{1}=10$ ways to do this. Now choose any four chips from the remaining 130 working chips, there are $\binom{130}{4}$ ways for this, so in total there are $$ \binom{10}{1} \binom{130}{4} = 10 * 11358880 = 113,588,800$$ possible choices.