Example 14 In a test with a full score of 100 points, all students' scores are non-negative integers. Let the scores of $n$ students be $a_{1}, a_{2}, \cdots, a_{n}$, where the number of students scoring at least $k$ points is $b_{k}$ $(1 \leqslant k \leqslant 100)$. Let $M=$ $a_{1}+a_{2}+\cdots+a_{n}, N=b_{1}+b_{2}+\cdots+b_{100}$, then the size relationship between $M$ and $N$ is ( ).
A. $M>N$
B. $M<N$
C. $M=N$
D. Cannot be determined
Solution: Choose C. Reason: On one hand, a student with a score of $k$ contributes $k$ to $M$. On the other hand, a student with a score of $k$ is counted exactly once in each of $b_{1}, b_{2}, \cdots, b_{k}$, but not in $b_{k+1}, b_{k+2}, \cdots, b_{100}$. Therefore, a student with a score of $k$ also contributes $k$ to $N$ for $1 \leqslant k \leqslant 100$. Thus, $M=N$, so the answer is C.