The Spirit Club recently got a call from the school executive council, asking them to make a committee. They now have $10$ volunteers and will need to choose a three-person committee. They need to calculate $_{10}C_3$, the number of possible committees, but they have no calculator with them.   

1. Al had a thought: “I remember that the number of three-person committees you can select from $9$ volunteers is $84$. This would be the number of committees the tenth volunteer will NOT be on because he or she would be selected from the other nine members.” How would you calculate this number?

   Answer (a):

   $_9C_3$

2. “I also know how many committees the new volunteer WILL be on because I can just add his name to each of the two-member committees I had already selected from $9$.” How many possible committees would include the tenth volunteer? 

   Partial Answer (b):

    $_9C_2=$?

3. Without using a calculator (and without multiplying or dividing), explain how Al used parts (a) and (b) to calculate $_{10}C_3$.

   Hint (c):

   Add your answers from parts (a) and (b).

4. How is this related to Pascal’s Triangle?

   Hint (d):

   How do you get from one row to the next when writing out Pascal's Triangle?