[Combinatorics]
Problem: Let $T=T N Y W R$. Kay has $T+1$ different colors of fingernail polish. Compute the number of ways that Kay can paint the five fingernails on her left hand by using at least three colors and such that no two consecutive fingernails have the same color.

Solution: There are $T+1$ possible colors for the first nail. Each remaining nail may be any color except that of the preceding nail, that is, there are $T$ possible colors. Thus, using at least two colors, there are $(T+1) T^{4}$ possible colorings. The problem requires that at least three colors be used, so we must subtract the number of colorings that use only two colors. As before, there are $T+1$ possible colors for the first nail and $T$ colors for the second. With only two colors, there are no remaining choices; the colors simply alternate. The answer is therefore $(T+1) T^{4}-(T+1) T$, and with $T=10$, this expression is equal to $110000-110=\mathbf{1 0 9 8 9 0}$.

Final answer: 109890