premutation fundamental problem?

Question

On a 15-item test, the first 5 items have 4 choices each, the next 5 items have 3 choices each, and the last 5 items are true or false. If Joe answers items 2, 7, and 10 correctly and guesses on all the others, how many different ways can he complete the test?

PLEASE HELP. I POSTED this ? before and i didnt get any helpful answers!!!

Answer 1

For Item 1-5,
there are 4 choices, Joe can answer it in 20 different ways

5 items x 4 choices/item = 20 choices

For item 6-10,
there are 3 choices, Joe can answer it in 15 different ways

5 items times 3 choices/item = 15 choices

For item 11-15,
there are 2 choices, Joe can answer it in 10 different ways

5 items times 2 choices/item = 10 choices

Since Joe answered item 2, 7 and 10 correctly, the number of different ways he can answer the others is 35.

Total choices = 45
Less choices for item 2 = 4
Less choices for item 7 = 3
Less choices for item 10 = 3

45 - 4 - 3 - 3 = 35

Answer 2

Questions 1-5 have 4 choices each
Questions 6-10 have 3 choices each
Questions 11-15 have 2 choices each (true or false)

Question 2, 7, and 10 are answered correct, so there's only 1 way to complete those:

To get the answer, multiply the number of possibilities for each question toegether. Questions 2, 7, and 10 have only 1 possibility.

Thus, the answer is:

4 x 1 x 4 x 4 x 4 x 3 x 1 x 3 x 3 x 1 x 2 x 2 x 2 x 2 x 2 = 221,184

Answer 3

This is an application of the multiplication rule of counting outcomes. The total number of outcomes for a procedure is the product of the numbers of outcomes for each stage.

The first 5 questions have 4 possible outcomes each, except #2, which has only one possible outcome since it was answered correctly.
The next 5 questions have 3 possible outcomes each, except for questions #7 and #10, which are answered correctly.
The last 5 questions have 2 possible choices each (T/F).

So, the total number of ways that this exam can be completed is:
4*1*4*4*4*3*1*3*3*1*2*2*2*2*2 = 221184

Answer 4

I assume that when you say he answered three questions correctly you mean he wasn't guessing on those questions because he knew. He may have answered additional questions correctly but only because of a lucky guess.

In that case:

For the first 5 questions he guessed on 4
For the second 5 questions he guessed on 3
For the last 5 questions he guessed on all 5

Number of ways to complete the test is:

(1*4^4)*(1*1*3^3)*(2^5) = 221,184