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This makes the calculation by hand fairly simple We will solve some interesting problems after stating and proving some facts about these binomial coef cients It is also true that n n = r n r An easy way to see this is to notice that if r objects are selected from n objects, then n r objects are left unselected Every change in an item selected produces a change in the set of items that are not selected, and so the equality follows It is also true that n = n 1 + n 1 To prove this, suppose you are a r r 1 r member of a group of n people and that a committee of size r is to be selected Either you are on the committee or you are not on the committee If you are selected for the committee, there are n 1 further choices to be made There are n 1 committees r 1 r that do not include you So

void P u s h ( T v a l u e ) { Node n new Node ( ) ; n m_va l u e = v a l u e ; Node h; do h = m_head ; n m_next = h ; while ( I nterloc ked Com p a re E x c hange ( ref m_hea d , n, h) } != h);

You may need to look carefully at that code to convince yourself that it's right We construct a new Node object to hold the value being pushed and immediately enter a d o - w h i l e loop Inside this loop we read the m_h e a d field into a local variable h We then set the new node's next pointer t o h Notice that although this value could b e out-of-date right away, setting it is safe; because we've not yet made the new node n publicly visible yet, no other thread can possibly see this value We then try to make it visible with an I nt e r l o c ked Compa r e E x c h a n g e We replace the current reference in

The Form 940 is shown in Exhibit 113 To fill out the form, follow these steps: 1 Identification section If the form is not preaddressed, fill in the business s address and employer identification number (EIN) If there is no EIN, apply for one with the Form SS-4 If no EIN has been received from the government by the time the form is due to be filed, print Applied For and the date of application in this area 2 Questions A-C If the answers to all three questions posed in this section are Yes, then one may file the simpler Form 940-EZ instead Any No answers require completion of the Form 940 The questions match the logic steps noted earlier in Exhibit 112 3 Amended or final return There are two check-off boxes immediately above the Part I section Check off the first box if there will be no future FUTA

it fails, we go back and try again The m_h e a d variable is marked v o l a t i l e to ensure w e properly reread i t during the next iteration o f the loop The Pop operation works similarly

Many other facts are known about the numbers n , which are also called r binomial coef cients because they occur in the binomial theorem The binomial theorem states that

T Pop O { Node n ; do { n m_hea d ; if ( n n U l l ) t h row n e w E x c eption ( " st a c k empty " ) ;

== =

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} w h i l e ( I nt e rlocked Compa r e E x c h a nge ( ref m_h e a d , n m_next , n ) ! = n ) j ret u r n n m_va l u e j }

This is fairly easy to see Consider (a + b)n = (a + b) (a + b) (a + b), where there are n factors on the right-hand side To nd the product (a + b)n , we must choose either a or b from each of the factors on the right-hand side There are n i ways to select i b s (and hence n i a s) The product (a + b)n consists of the sum of all such terms Many other facts are known about the binomial coef cients We cannot explore all these here, but we will show an application, among others, to acceptance sampling