A simpler way to think of the approximation is that 24! multiples together all fifteen numbers from 10 to 24, all of which add at least another power of 10, and

A simpler way to think of the approximation is that 24! multiples together all fifteen numbers from 10 to 24, all of which add at least another power of 10, and you get approximately another nine powers of ten from multiplying in 1 through 9. You end up around 6.2e23 which rounds up to the approximation. Going up to the next factorial bumps it up (approximately) another power of ten since you are multiplying by 2.5 and 10. In reality it’s around 1.5e25 which is also a reasonable approximation. Over simplification: a number times 25 is closer to the number times 10 than it is to the number times 100.