# Problem 3 of 3

At a concert, the program included three vocal presentation and two piano presentations.

(a) In how many ways can the program be arranged so that it includes all the presentations?

(b) In how many ways can the program be arranged if piano and vocal presentations alternate?

(c) If the order of the presenations was randomly selected, what is the probability that piano and vocal presentations would alternate?

(d) You are giving one of the piano presenations. If the order of the presentations is randomly arranged, what is the probability that you will go first?

**Solution**

a)The way on how u would calculate this problem is as follows:

You should use the factorial notation method which is n!=n*(n-1)(n-2)(n-3)......3*2*1

When you get 0!=1.

So for this question the first number in the factorial sequence would be 5 because there are 5 different presentations and if it doesnt matter for which one goes first then there are 5 alternatives for the first one to go on stage so the whole thing would like the following formula:

5!=5*4*3*2*1=120 different ways they can be arranged.

To do this on your TVM solver(calculator)use the following steps:

5!=> 6,MATH,<,4,and then enter

4=(!) sign

<=(PRB) button

SO it will looke like this on the screen 5! 120

b) This problem was a little more complicated to solve because u have to take into consideration that in order for the 2 different presentations to completley alternate they can only be placed in one order that is:

P=Piano

V=Vocal

V*P*V*P*V=Total alteration

Since there are only 3 differnt vocal acts "3" would be the first number to go into the Vocal slot and then in the second slot it would be a "2" because there are only 2 piano acts and any 2 can be first.

So the Formula would look like the folowing:

3*2*2*1*1=12 different ways in which the performances can alternate at choosing and eliminating.

c)In order to do this problem you have to take into consideration that these performances are being choosen at random and for that reason it like picking a number out of a hat and putting it bac if you catch my drift.

So the first number in this formula would be 3 again because as i said before they can only alternate in one way and thats V*P*V*P*V, so the next number as you would imagine is 2,and then this is where the random selection comes in so since we can use any of the other perormances again in the tird slot we will use 3 for vocal again so the formual would be as follows:

3*2*3*2*3=108 different ways the performances can be alternated randomly.

d)

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