Subsets of Combinations

Date: 10/05/2003 at 19:38:19
From: Tyler
Subject: Subsets

How many subsets of size 10 of the numbers 1, 2, 3, ..., 20 have 5 as
the smallest element?

Date: 10/05/2003 at 20:23:14
From: Doctor Luis
Subject: Re: Subsets

Hi Tyler,

There are 

  C(20,10) = 20*19*18*...*10 / 10! 

           = 184756

subsets with 10 elements because you are choosing 10 elements out of
20.  Now, how many of these contain 5 as an element?

If you choose 5, then the other 9 elements in your subset are chosen
from 19 possibilities, hence there are 

  C(19,9) = 92378 

subsets which contain 5.

However, 5 is supposed to be the smallest element.  So, the numbers 1,
2, 3, and 4 are forbidden.  In reality, you don't have 19 choices to
fill those 9 extra spots in your subset.  You have only 15 choices: 
 
  6, 7, ..., 19, 20

So, there are C(15,9) choices for the other 9 elements.  How many
would that be?  I'll leave that for you to calculate :)

Get it?

I hope this helped! Let us know if you have any more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/