Given 100 students who are to be assigned a grade 1 to 5, what is the number of possible assignments if we are interested in knowing what grade each student received. Furthermore, we are given the constraints that at least 30 students receive a grade of 4.
The way I approached this problem was by saying that since in all grade distributions at least 30 students must achieve a grade of 4 then we can "reserve" those using $100\choose 30$. This leaves us with 70 students that can be assigned grades using $5^{70}$. The final result would then be given by
$${100\choose 30}\cdot{5^{70}}$$
However, the actual solution is $$\sum_{k=30}^{100}{100\choose k}{\cdot}{4^{100-k}}$$
which also makes sense to me. I just can't seem to make out how exactly the two constructs differ and what my version is actually representing.
Best Answer
let's use smaller numbers for intuition. Assume two grade levels and 3 students. Assume again at least one student gets lower grade. So you have these scenarios
$\{1,2,2\}, \{2,1,2\}, \{2,2,1\}, \{1,1,2\},\{1,2,1\},\{2,1,1\}, \{1,1,1\}$
using your way will give $ {3 \choose 1} 2^2 = 12$, which is clearly wrong.
Using the proposed solution: $$\sum_{k=1}^{3}{3\choose k}{1^{3-k}}= 3 + 3 + 1 = 7$$