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Please write the probability formula for each one.

1. Twenty five scientists are seeking financial support from the National Science Foundation. Find the number of ways in which the NSF panel can select four of the twenty five submitted proposals ranking them as first second , third , and fourth.

2. A basket contains 6 red apples and 9 green apples . A sample of 5 apples is drawn. Find the probability that the sample contains
a. Five red apples?
b. Two red and three green apples?
c. at lest one red apple?

3. If two cards are drawn fro0mthe standard deck without replacement, find the probability that :
a. The first card is red, and the second is black
b. Both cards are aces

4. There are seven cards with digits 0,1, 2, 3, 4, 5, and 6. A child takes four of the cards and places them in some order. What is the probability that he obtains a four-digit number less than 3265?

Please include the formula for each part of these question and show all steps. Thank you

Solution Preview

(1) The number of ways in which 4 selections can be done out of 25 is C(25, 4) = 12650

Any one of the 4 selected submissions can be ranked as First. This can happen in 4 ways.

Any one of the remaining 3 submissions can be ranked as Second. This can happen in 3 ways.

Similarly the other two rankings can be given in 2 ways and 1 way respectively.

Therefore, by Fundamental Counting Principle, total number of ways of selecting 4 submissions and ranking them is

12650 * 4 * 3 * 2 * 1 = 303600.

(2) Total number of apples in the basket = 6 + 9 = 15

5 apples can be drawn out of ...

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