1. how many injective maps?

In a homework assignment I was given the following question

Make a list of all the injective maps f : {1,2,3} → {1,2,3,4}.
Show that none is bijective.

After writing down all the possible maps I got a total of 24 injective maps. For the second part of the question I was asked

How many injective maps are there?
f : {1,2,3,4,5,6,7,8} → {1,2,3,4,5,6,7,8,9,10}

Looking over my notes we never got a general formula on how to calculate how any injective maps would be posible, and if I were to write out each possible injective map by hand it would probably take a really long time and I would probably mess up along the way. I would think if there is a formula on how to calculate the number of injective maps for the above problem it would be 10! / 8! but that would only be equal to 90 injective maps which is too few. Any help would be appreciated.

2. How many injective maps are there?
f : {1,2,3,4,5,6,7,8} → {1,2,3,4,5,6,7,8,9,10}
f(1) can be any element of the codomain {1,2,3,4,5,6,7,8,9,10}: 10 choices. Next, f(2) is not equal to f(1), so f(2) can be any element of {1,2,3,4,5,6,7,8,9,10} except for f(1): 9 choices. Similarly, there are 8 choices for f(3), ..., and 3 choices for f(8).