Probability problem

First compute the probability that a question will not have been used at all in the first four exams. The probability that a question will not be used for any given test is 300/350 or 6/7. The probability that you can escape four tests is that number to the fourth power = 54%.

There may be a faster method for the second part, but I did it by computing the chance of getting zero previous questions, one previous question, etc until I got up to nine and then just added those values together. The probability of getting exactly n previously used questions is [50! / (50-n)!/ n!] P^(50-n) (1-P)^n.

Here are the probabilities for each value
0 4.07983E-14
1 1.73928E-12
2 3.63323E-11
3 4.95644E-10
4 4.96552E-09
5 3.89502E-08
6 2.49074E-07
7 1.33488E-06
8 6.11753E-06
9 2.43411E-05
10 8.50906E-05
11 0.000263819
12 0.00073105
13 0.001821983
14 0.004105588
15 0.008401249
16 0.015669285
17 0.026719999
18 0.04176718
19 0.059977562
20 0.079264328
21 0.096546608
22 0.108509963
23 0.112630779
24 0.108035599
25 0.095798233
26 0.078538289
27 0.059523189
28 0.041688212
29 0.026964678
30 0.016093502
31 0.008852708
32 0.004481641
33 0.002084265
34 0.000888546
35 0.000346329
36 0.000123037
37 3.96935E-05
38 1.15781E-05
39 3.03746E-06
40 7.12197E-07
41 1.48106E-07
42 2.70598E-08
43 4.29242E-09
44 5.82244E-10
45 6.61914E-11
46 6.13438E-12
47 4.45133E-13
48 2.37207E-14
49 8.25503E-16
50 1.40769E-17

The sum probability of having 10 or more questions is 0.999967913 so we'll just call that 100%.

The problem statement is a bit ambiguous. I interpreted the problem to be:
After selecting questions for five exams, what is the probability that ten or more questions appeared on at least two exams?
But this method seems to be trying to solve this problem:
When making the selections for the fifth exam, what is the probablity that ten or more of the current selections will have been selected previously?

In any case, this is wrong. The questions for each test would be sampled without replacement. The formula works for sampling with replacement. Testing the formula on a small case demonstrates this. For instance:
Question bank for exam has 7 questions from which 2 are selected at random for each exam. By the time of the third exam, what is the probability that n questions have been used more than once?