Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 1 / d / Solution

Let $x_i$ be the number in box $i$ and put $y_i=x_i-1$. We count integer compositions satisfying $y_1+\cdots +y_4=6$ and $0\le y_i\le 4$. Without the upper bounds, stars and bars gives $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$. If a specified $y_i\ge 5$, subtract five from it; the remaining nonnegative integer variables sum to one, giving $\left(\genfrac{}{}{0ex}{}{4}{3}\right)=4$ possibilities. Two violations cannot occur because they would require a sum at least ten. The inclusion-exclusion principle therefore givesThe boxes are distinguished by their positions in the composition; individual objects are not distinguished.