I was wondering if anyone could help me answer these questions? Thank you so muc

Question

I was wondering if anyone could help me answer these questions? Thank you so much!

1. Find the number of ways a four-digit number could be made using the digits {1,2,3,4,5} if the digits can be repeated.

2. Find the number of ways a four-digit number could be made using the digits {1,2,3,4,5} if the digits cannot be repeated.

3. There are 15 members in the High School Service Club. The club is to send 4 representatives to the State Conference. How many different ways are there to select a group of four students to attend the conference?

4. The High School Service Club described in Excercise 3 has decided to work on a community project. Before beginning, it needs to decide which of its members will be the project coordinator, which will be the site director, and which will be the financial manager. How many ways are there to fill these positions?

5. Determine the total number of five-card hands that can be drawn from a deck of 52 cards.

6. A track has 8 lanes, and 8 runners need to be assigned to their own lane for an event. How many ways can they be assigned?

7. A ship has 9 different signal flags available. To form a signal, 3 flags are flown one above another along a vertical pole. How many different signals can be made?

8. Kyra has missed several of her biology classes and has not even read the material for the days she missed. She has just received a 10-question True/False quiz. What is the probability that she got all of them right?

9. In the Illinois Lottery game Little Lotto, 5 balls are drawn without replacement from an urn containing balls numbered 1-30. To win, all 5 of the numbers on a player's ticket must match all 5 of the numbers drawn from the urn. Note that the order in which the balls are selected does not matter. What is the probability of a ticket holding the winning numbers?

10. On a television game show, a contestant is shown a set of 5 prizes and a set of 5 price tags. If he matches all of the price tags to their correct prizes, then he wins all of the prizes; otherwise, he wins nothing. What is the probability that he wins by placing the price tags with the prizes completely at random?

• In his article, "American Football" (published in Statistics in Sports ), Iowa State University statistician, Hal Stern, evaluates winning strategies of teams in the NFL. In a section on estimating the probability of winning a game, associated with certain outcomes (i.e. running play, short pass plays, long pass plays). The following table gives the probability distribution for the yardage gained, X, on a running play. Note that a negative gain represents a loss of yards on the play.

1. Verify that X has a well-defined probability distribution.

2. Find the probability of gaining 10 yards or more on a running play.

3. Find the probability of losing yardage on a running play.

4. Find and interpret the expected value, or mean of the distribution.

5. Determine the standard deviation of the distribution.

X -4 -2 -1 0 1 2 3 4 6 8 10 15 30 50 99 P(x) .020 .060 .070 .150 .130 .110 .090 .070 .090 .060 .050 .085 .010 .004 .001

Explanation / Answer

1)

in this case we have 5 choices and 4 places to fill with repetition, so it can be explained as

1st place of 4 digit number can be filled in 5 ways, 2nd place is also filled in 5 ways since repetition is allowed. similarly, 3rd and 4th places are also filled in 5 ways

so total number of 4 digit number is 5*5*5*5=625

2)

in this case we have 5 choices and 4 places to fill without repetition, so it can be explained as

1st place of 4 digit number can be filled in 5 ways, 2nd place is also filled in 4 ways since repetition is not allowed and 1 digit out of 5 is utilized in 1st place. Similarly, 3rd place will be filled in 3 ways and 4th place is filled in 2 ways

so total number of 4 digit number is 5*4*3*2=120

3)

group of size 4 need to be formed by selecting 4 members from 15 members.

so, number of ways =15C4=15!/4!(15-4)!=15!/4!/11!=(15*14*13*12*11!)/(4*3*2*1*11!)=1365

4)

since it is given that for all the three positions 3 individuals will be chosen

if at first project coordinator, at 2nd site director, and at 3rd financial manager are selected then the total number of ways is 15C1 *14C1*13C1 =15*14*13=2730

similarly, if at first site director, at 2nd project coordinator, and at 3rd financial manager are selected then the total number of ways is 15C1 *14C1*13C1 =15*14*13=2730

if at first financial manager, at 2nd project coordinator, and at 3rd site director are selected then the total number of ways is 15C1 *14C1*13C1 =15*14*13=2730

so in total 3! ways are there in which these 3 positions are to be filled

so total number of ways =2730*3!=16380

5)

total number of 5 card hands = 52C5=52!/5!*(52-5)!=2598960

6)

1st individual is assigned to 1 lane in 8 ways 2nd individual is assigned to 1 lane in 7 ways,....... 8th individual is assigned to 1 lane in 1 way

so total number of ways

8*7*....2=8!=40320

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