Suppose there are $10$ pairs of chocolates and each pair is distinct. To find the number of ways I can choose $5$ chocolates from $20$ such that none of them will be the same pair.
I tried the problem with the following argument:
Consider the pairs as blocks. So practically we choose $5$ chocolates from $10$ in $\binom{10}{5}$ ways. Since for each block, there are two identical chocolates we can choose it in $\binom{2}{1}$ ways. So the total number of ways is $$\binom{10}{5} \times \binom{2}{1}^5.$$
Is my argument correct?