How many subsets of $K=\{1,2,3,...,20\}$ of 4 elements can we create that contain at least one of the numbers $1,2,3,4,5$?
I think that there are generally $20^{4}=160,000$ subsets of $4$ elements each and the answer to the problem is $5^{4}=625$, however I am not really sure.
The number of subsets of $\{1,2,\ldots,20\}$ consisting of $4$ elements is $\binom{20}{4}.$
The ones that are not allowed are exactly the subsets of $4$ elements in $\{6,7,\ldots,20\}$. There are $\binom{15}{4}$ such subsets, so the answer is $$\binom{20}{4}-\binom{15}{4}.$$