Error in proof by strong induction

Question

$ \textbf{Question:} $ If an experiment can be performed in $ k $ steps such that the first step can be done in $ n_{1} $ ways, the second step can be done in $ n_{2} $ ways, ..., and the last step can be done in $ n_{k} $ ways, then the total number of ways to perform this experiment is $ n_{1} + n_{2} + \dots + n_{k}. $

Here is the proof using strong induction for this $ \textbf{false} $ statement, can someone help me find the error in the proof?

$ \textbf{Proof:} $ Let $ P(k) $ be the statement that "if an experiment can be performed in $ k $ steps such that the first step can be done in $ n_{1} $ ways, the second step can be done in $ n_{2} $ ways, ..., and the last step can be done in $ n_{k} $ ways, then the total number of ways to perform this experiment is $ n_{1} + n_{2} + \dots + n_{k}. $"

Base case: If the experiment only has $ 1 $ step, then there is $ n_{1} $ ways to do it, so $ P(1) $ is true. Now suppose that $ P(1),P(2), \dots, P(k) $ is true for $ k > 1. $ Consider $ P(k + 1), $ since if the experiment has $ k + 1 $ steps, it can be done by first doing the first $ k $ steps following by the $ k + 1 $ steps, thus the total number of possible ways to do the experiment is $ (n_{1} + n_{2} + \dots + n_{k}) + n_{k + 1} $ (since $ P(2) $ is true). Hence $ P(k + 1) $ is true as well.

Answer 1

The induction step fails. Think about $P(2)$. You have $n_1$ ways to do the first step. For each of those, you have $n_2$ ways to do the second step. The proof misses the point that you then have $n_1n_2$ instead of $n_1+n_2$ ways to do the first two steps.

Answer 2

The induction step is erroneous because the inference used is unjustified (and, in fact, fallacious).

From the universal principle of counting: the number of ways to perform two subtasks in serial is the product of the ways to perform each of the subtasks.

Thus if the first $k$ steps can be preformed in $\sum_{j=1}^k n_j$ ways, and the next step can be performed in $n_{k+1}$ ways, then the first $k+1$ steps of the experiment can be performed in $\bbox[lightpink,0.25ex]{~n_{k+1}\sum_{j=1}^k n_j~}$ ways.

This is clearly not generally $~\sum_{j=1}^{k+1}n_j~$ as you would require for the prof to be sound.

tl;dr $P(2)$ is not true.