You might want to draw up a "sample space" showing all of the possibilities and then counting how many satisfy the "sum is odd " criterion.
Start with Bill's cards: {4,5,6}. Choose the 4 (that's arbitrary). Add that 4 to Ben's cards, one by one:
4+4 = 8 (even)
4+5 = 9 (odd)
4+6 = 10 (even
Now choose Bill's 5 and add that, one by one, to each of Ben's;
5+4=9 (odd)
5+5 = 10 (even)
5+6 = 11 (odd)
Now choose Bill's 6 and add that to each of Ben's, one by one:
6+4=10 (even)
6+5=11 (odd)
6+6 = 12 (even)
As you can see, there are a total of 9 possible outcomes. How many of these outcomes are odd? I count 4.
Thus, the chances of satisfying the "odd sum" criterion is 4/9 (answer)
