Permutations problem?

Question

i have a problem with several of the excersices in my book.

1) a true false test consisting of 20 questions can be marked in 1,048,576 different ways. In how many ways can each question be marked true or false so that at leat 17 are right?

==>the answer is 1351 , but how?

2) In a primary election, there are 4 candidates for mayor, 5 candidates for city treasurer, and 2 candidates for county attorney.

In how many ways can a person vote if he excercises his option of not voting for a candidate for any or all of these offices?

==> the answer is 90 , how?

Note , did try these exercises ,after exausting all possible formulas, ideas combinations

thanks in advance

Answer 1

1)this the answer of your first question.

            20!/17!3!+20!/18!2!+20!/19! + 1              =20*19*18/6+20*19/2+20=1140+190+20+1=135...

you have four possible position

1)you have 17 right

2)you have 18 right

3)you have 19 right

4)you have 20 right

well for position 1 there are 20!/17!3! possible ways

for position 2 there are 20!/18!2! possible ways

for positon 3 there are 20!/19! possible ways

and for the last position there is 1 possible way.

Answer 2

1) combination, not permutations:

20C17 + 20C18 + 20C19 +20C20 = ?

That is, the number of ways 17 can be chosen from 20, plus the number of ways 18 can be chosen from 20,...

[Hint: there is no more point in figuring out combinations by hand as there is figuring out square roots by hand. Use a tool!  It is far more important that you understand the meanings of combinations and permutations and how to apply them.]

2) Since there are 4 candidates for mayor and you can cast no vote, you actually have 5 choices. You have 6 for treasurer, and 3 for county attorney:

5 * 6 * 3 = ?