Finite math help please

Question

1. A state Motor Vehicle Department requires learners to pass a written test on the motor vehicle laws of the state. The exam consists of twenty true-or-false questions, of which eighteen must be answered correctly to qualify for a permit. In how many different ways can a learner who answers all the questions on the exam qualify for a permit? 

Answer 1

18 correct --> 20 choose 18 = 20! / [ 18! 2!] = 20*19/2 = 10*19 = 190

19 correct --> 20 choose 19 = 20 / [ 19! 1!] = 20

20 correct --> 20 choose 20 = 20! / ( 20! 0!) = 1

190+20+1 = 211

Answer 2

ways to pass include ways to get 0 or 1 or 2 questions wrong

With 20 questions there is just one way to get 0 wrong.

20 ways to get 1 wrong

to get 2 wrong includes missing the 1st and one of the 19, that's 19 ways

or miss the 2nd and one of the remaining 18, that's 18 ways

miss the 3rd and one of 17 for 17 ways

miss the 4th and one of 16 for 16 ways

miss he 5th and one of 15 for 15 ways

....

miss the 19th and 1 of 1 remaining for 1 way

that sums up to 1+20 + 19+ 18 +17 + 16 + 15 .... + 3+ 2 + 1

sum of 1 through 20 = (20)(21/2 = 210

210 + 1 = 211 ways to pass the test, getting at least 18 right and no more than 2 wrong.