Key idea. Find the smallest possible total if every cage population is different.

1. Suppose all 14 cages contain different numbers of rabbits. List those numbers from smallest to largest.

2. The first number is at least 0, the second at least 1, the third at least 2, and so on. The fourteenth is at least 13.

3. The total must therefore be at least

   \[0+1+2+\cdots+13=7\cdot13=91.\]

   To get 91, pair the first and last numbers, then the second and second-last: there are seven pairs, each summing to 13.

4. There are only 90 rabbits. So the assumption that every cage population is different cannot be true.

Remember. When numbers must be distinct, ordering them gives a lower bound on their sum.