How Do You Solve These Probability and Combinatorics Problems?

[wmosley]

need your assitance!

Anyone know how to do the problems below...



1)

There are 5 rotten plms in a crate of 25 plums. How many samples of 4 of the 25 plums contain



A) Only good plums?

B) Three good plums and 1 rotten plum?

C) One or more rotten plums





2)

An admission test given by a University contains 10 True-false questions. Eight or more of the questions must be answered correctly in order to be admitted.



A) How many different ways can the answer sheet be filled out?

B) How many different ways can the answer sheet be filled out so that 8 or more questions are answered correctly?

 

[Tide]

I'm sure LOTS of people can do those problems - the real question is what have YOU done on them so far? :-)

Also, this should be in the homework section.

 

[recon]

For question 1:
You know that you have 5 rotten plums and 20 good plums.

(A) asks you the number of ways that you can choose 4 good plums from 20 good plums. In case you cannot tell the difference between a permutation and a combination, let me point out that this is a combination. Use the formula ⁿC_r for finding the answer.

(B) is a bit more tricky. It asks you the number of ways you can choose 1 rotten plum from 5 rotten plums, and 3 good plums from 20 good plums. Determine ⁿC_r for both and multiply them together.

(C) can be done using a short-cut method. Think of the number of ways you can choose 4 plums from 25 plums. Then, think of the number of ways 4 GOOD plums can be chosen. If you know sets well, the answer will then come easily to you.

 

[recon]

For Question 2 Hints:
A) Consider for a moment the number of ways in which 2 true/false questions can be answered. There are 2 ways in which you can choose the answer for the first question, and another 2 ways in which you can choose the answer to the second question. Altogether, there are 2 X 2 ways in which the two question could have been answered.

B) There are three possibilities here: (i)there are 2 wrong answers, (ii) there is 1 wrong answer, and (iii) there is no wrong answer. For (i) you have to ask yourself, "In how many ways can I answer 2 questions wrong and get 8 correct?". Here's are some possible combinations for (i):
WWRRRRRRRR
WRWRRRRRRR
WRRWRRRRRR
WRRRWRRRRR
WRRRRWRRRR
WRRRRRWRRR
WRRRRRRWRR
WRRRRRRRWR
WRRRRRRRRW

Can you see the pattern here? It continues below.

RWWRRRRRRR
RWRWRRRRRR
RWRRWRRRRR
RWRRRWRRRR...

So, if you understood all that, you'll now see that there are altogether 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 ways under which condition (i) is fulfilled.

There are 10 ways in which condition (ii) can be fulfilled. Have a look:

WRRRRRRRRR
RWRRRRRRRR
RRWRRRRRRR...

And only 1 way in which condition (iii) is met, i.e. RRRRRRRRRR.