Question

In one year, three awards (research, teaching, and service) will be given to a class of 25 graduate students. If each student can receiveat most one award, how many possible selections are there? Hint: this is a choice problemwhere order matters and repetition is not allowed

Answer 1

Answer:

Step-by-step explanation:

Out of 7, 3 people need to be selected and arranged (into 3 different positions: President, VP, Sec)

So there are 2 ways to do it: 7 * 6* 5 (you say, select the president in 7 ways, now select VP in 6 ways and then select Sec in 5 ways) = 210

or you can use the permutation formula nPr such that nPr = n!/(n-r)!. nPr helps you select r people out of n people AND arrange those r people.

Above, we used 7P3 = 7!/(7 - 3)! = 7!/4! = 7*6*5 = 210

Now I am assuming that your question is why the formula is n!/(n-r)!

Say you have n people and you want to arrange them. You can do it in n! ways, right? Just our basic counting principle. Say there at 7 people and you want to arrange all 7 in 7 spots. You can do it in 7! ways ( = 7*6*5*4*3*2*1). 7 ways to fill the first spot, 6 ways to fill the second. 5 ways to fill the third, 4 ways to fill the fourth etc.

Now what if you have only 3 spots? You have to fill 3 only. You can do it in 7*6*5 ways. What about the rest of the 7-3 = 4 spots? (which is n - r) You have to ignore them. So if you do arrange people in 7 spots by using 7! in the numerator, you must divide out the extra n - r spots i.e. 4!. That is the reason you divide by 4! here. ..

If it will help you then plz give it a brainlist award. . . And vote 2hh. . . ❤

Answer 2

Answer:

The required answer is: $13,800$

Step-by-step explanation:

Because order matters, this is a permutation rather than a combination.

This is why:

• Let's imagine, purely arbitrary, that Abby, Bob, and Charlie are the three students selected to receive the honor. The three of them should receive each award in the same order; otherwise, they would each receive a different award. For instance, it would be very different if Abby received the teaching award, Bob received the service award, and Charlie received the research award if Abby received the research award, Bob received the teaching award, and Charlie received the service award.

• We, therefore, have a pool of 25 candidates to choose from for the first award. Since the problem states that each student can only win one award, we have 24 options after we chose that person. 23, then, and so forth.

• There are three candidates for the awards, hence there are:

$25$×$24$×$23=13,800$ possible permutations. This is our last response since order matters (please see the justification above).

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