Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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Permutations containing a given subsequence

Let $f(n)$ denote the number of $4n$-long strings formed from $2n$ a's and $2n$ b's, such that the string contains, as a (possibly non-consecutive) subsequence, a pattern containing $n$ a's and $n$ ...