We derive partial solutions for a recently-posed problem of the enumeration of restricted words. We obtain several explicit formulas in which the number of restricted words is expressed in terms of the binomial coefficients. These results establish relations between the partial Bell polynomials and the binomial coefficients. In particular, we link the r-step Fibonacci numbers, the binomial coefficients, and the partitions of a positive integer into at most r parts. Also, we prove that several well-known classes of integers can be interpreted in terms of compositions. We finish the paper with an extension of a recent result about Euler-type identities for integer compositions.