AoPS Counting and Probabilty Help . . . Please?

[Jenny in Florida]

My son is trying to do one of the challenge problems, 2.29 in Intro to Counting and Probability. He's stuck, and neither my husband nor I are able to figure out how to unstick him. I'd appreciate any help anyone could offer?

 

The n members of a committee and numbered 1 through n. One of the members is designated as "the Grand Pooh-bah." The n members sit in a row of n chairs, but no member greater than the Grand Pooh-bah may sit in the seat to the immediate right of the Grand Pooh-bah. Suppose that the Grand Pooh-bah is member number p, where 1 is less than or equal to p is less than or equal to n. Find a formula, in terms of n and p, for the number of ways for the committee to sit.

 

Using hints from the solutions manual, we got as far as:

 

(n-1)! + (n-1)(p-1)(n-2)!

 

But we're supposed to simplify, and I'm completely confused. I know what the solutions manual says the answer should be, but we can't figure out how to get there.

 

I know there are many smart and mathy folks here. So, help please?

[Kathy in Richmond]

Yay for fun math!

 

I followed the solution up to your result, and I agree so far. Here's what I'd do to simplify it.

 

(n-1)! + (n-1) (p-1) (n-2)! =

 

First note that (n-1)! = (n-1) (n-2) (n-3) ... 1 = (n-1) (n-2)!, so

 

= (n-1)! + (p-1) (n-1)!

 

= [(n-1)!] [1 + (p-1)] by factoring out the common term

 

= (n-1)! [ p] by simplifying the last term

 

So I get p(n-1)! as the final answer.

[Jenny in Florida]

So I get p(n-1)! as the final answer.

 

That's what the book said, too. We just couldn't for the life of us figure out how to get there. Thank you so much!

[Kathy in Richmond]

You're welcome - ask any time...I love this stuff!:)