That's not really the right reasoning to use here. The number of cell divisions isn't 50 vs. 54, it's more like 2^50 vs. 2^54.
At some point the animal will exit the growth phase and reach a stable cell count and an elephant that reaches adulthood will just simply have more cells than a mouse. A 5,000kg elephant has a lot more cells that could develop cancer than a 0.02kg field mouse. And if that elephant lives 60 years instead of 1.5 for the mouse (let's just say for the sake of argument that cells divide once per year for replacement), that could be something like a 10,000,000 fold difference in the number of "cell-years" and cell divisions (at once per year) for something to go wrong and cause one of those cells to become cancerous.
"Peto noted that, in general, there is little relationship between cancer rates and the body size or age of animals. That is surprising: the cells of large-bodied or older animals should have divided many more times than those of smaller or younger ones, so should possess more random mutations predisposing them to cancer. Peto speculated that there might be an intrinsic biological mechanism that protects cells from cancer as they age and expand."
So, yeah, it seems like something important has to be going on. If a mouse can die of cancer at 1 year old, how can any elephants survive to 60?
Prostate cancer is not a huge risk because the Prostate is a massive number of cells. So, cell type matters and elephants for example don't have the same number of skin cells relative to body mass as humans.
Cancer is not a single mutation. So, generally the first cancer cell is a descendent of the first cell that had a dangerous mutation. Further, some cells are being constantly produced over a lifetime (skin, GI tract etc), so you can mostly ignore mussel fiber or fact cells when calculating cancer risks.
In that context there are divisions related to growth and divisions related to homeostasis. Tacking on 4 extra related to growth is relatively less important vs lifespan and other factors especially when you look at human cancer risks pre 44.
A mouse lives at most three years in captivity, in the wild they only live one year due to predation so the effectiveness of the anti cancer mechanism in mice could be seen as equivalent to those of the elephant! :) Perhaps another interpretation could be - one animal species doesn't evolve into a larger mass animal unless accompanied by more effective anticancer measures.
Generally cells get cancer after having processed a certain amount of energy. That means that some fly species will reach obvious senescence in a matter of double-digit hours whereas for humans it takes 60 years or so. But cell-for cell, the cells in those bodies do "about" the same in terms of watts that go through them (nanowatts in reality, of course).
This is related to age and cell count, but it's not the only factor. For instance when an animal becomes larger, the size of the animal goes up with the third power, while energy use only goes up with the second power. So the bigger an animal, the less individual cells can do.
The way to arrive at this insight is to imagine animals are balls. To about a factor 2 this is accurate. The energy use is limited by energy exchange with the outside world, ie, it's limited by the amount of skin they have, which is the surface of the ball. The amount of cells is related to the volume the animal occupies.
The net effect of this is that bigger animals live longer. Some details are different too. For instance, larger animals tend to have larger cells. So the cell count goes up, but not by as much you'd think (If humans had mouse-sized cells in their tissues we'd be on average 54cm).
At some point the animal will exit the growth phase and reach a stable cell count and an elephant that reaches adulthood will just simply have more cells than a mouse. A 5,000kg elephant has a lot more cells that could develop cancer than a 0.02kg field mouse. And if that elephant lives 60 years instead of 1.5 for the mouse (let's just say for the sake of argument that cells divide once per year for replacement), that could be something like a 10,000,000 fold difference in the number of "cell-years" and cell divisions (at once per year) for something to go wrong and cause one of those cells to become cancerous.
"Peto noted that, in general, there is little relationship between cancer rates and the body size or age of animals. That is surprising: the cells of large-bodied or older animals should have divided many more times than those of smaller or younger ones, so should possess more random mutations predisposing them to cancer. Peto speculated that there might be an intrinsic biological mechanism that protects cells from cancer as they age and expand."
So, yeah, it seems like something important has to be going on. If a mouse can die of cancer at 1 year old, how can any elephants survive to 60?