The probability of opening a cereal box containing a new toy given you have $k$ toys is $$ P(\text{new} | k) = \frac{6-k}{6}. $$ First, we want to find the expected number of cereal boxes needed to find a new toy given we have $k$ toys. You can either see that opening a cereal box containing a new toy given $k$ toys is a Bernoulli trial, or calculate the expectation directly by setting up the recursion relation $$ \text{E}[\text{new} | k] = 1 \cdot P(\text{new} | k) + \left(1 - P(\text{new} | k)\right)[ 1 + \text{E}[X] ] $$ to determine that the expected number of boxes we need to open is $$ \text{E}[\text{new} | k] = \frac{1}{P(\text{new} | k)}. $$ To calculate the expected number of boxes we need to get all six toys, we need to calculate $$ \text{E}[N] = \sum_{k=0}^5 \text{E}[\text{new} | k] $$ which evaluates to $$ \text{E}[N] = \frac{6}{6} + \frac{6}{5} + \frac{6}{4} + \frac{6}{3} + \frac{6}{2} + \frac{6}{1} = 14.7. $$