Counting problem involving picking delegates

[torquerotates]

Homework Statement

An organization of 100 members, 6 of whom are officers, plans to elect delegates to attend a convention. There are to be 2 delegates; one must be an officer and the other cannot be an officer. In addition, an alternate delegate, either an officer or not, will be elected and will attend if one of the regular delegates is unable to do so. How many different outcomes can this election have?

Homework Equations

The Attempt at a Solution

So, there are 6 officers to choose from and 94 non officers to choose from. If one of them are unable to be a delegate then there are 99 people to choose from(100-1 for the other person that cannot make it). So in all there should be 6*94*99=55836 different ways of picking delegates. But the back of the book gave 55272 ways. Is there something I did that was wrong?

 

[Shooting Star]

Homework Statement

An organization of 100 members, 6 of whom are officers, plans to elect delegates to attend a convention. There are to be 2 delegates; one must be an officer and the other cannot be an officer. In addition, an alternate delegate, either an officer or not, will be elected and will attend if one of the regular delegates is unable to do so. How many different outcomes can this election have?

Homework Equations

The Attempt at a Solution

So, there are 6 officers to choose from and 94 non officers to choose from. If one of them are unable to be a delegate then there are 99 people to choose from(100-1 for the other person that cannot make it). So in all there should be 6*94*99=55836 different ways of picking delegates. But the back of the book gave 55272 ways. Is there something I did that was wrong?

Two people are already out of the pool when you are choosing the third delegate...

 

[torquerotates]

Oh i see. The officer and the nonofficer that was chosen. makes sense