The Counting Principle

The Fundamental Counting Principle says: If there are $m$ ways to one thing and $n$ ways to do another, then there are $m * n$ ways to do both. Of course this idea can be extended to more then two events also.

In this video we look at what the Counting Principle is and see how to apply it in different situations. Examples include independent events (events whose outcomes do not affect each other) and dependent events (events for which the outcome of one affects the outcome of the other.)

Try these problems:

1.  To use an ATM at a certain bank, you must enter a $5$-digit code, using the digits $0-9$.  How many $5$-digit codes are there if you are allowed to repeat a number.
2. License plates in Dragon Land start with two captial letters followed by $3$ digits ($0-9$).  How many different combinations are there if the letters can not be the same, the first of the three digits can not be $0$, repetition of digits is allowed.

Solutions:

1.  Key facts:  5-digit code, using the digits 0-9, allowed to repeat a number. Fine out how many 5-digit codes are there.
   1. There 5 options:   ___ ___ ___ ___ ___
   2. There are 10 possibilities (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) for each of the 5 digits
   3. $10 * 10 * 10 * 10 * 10 =100,000$
2. Key facts: license plates have 2 capital letters that can not repeat followed by 1 digit that can’t be $0$ and 2 digit that can include $0$, the digits can repeat.
   1. There are 5 options:  ___ ___ ___ ___ ___
   2. There are 26 possibles letters for the first letter and 25 possibles letters for the second letter: $26 * 25 *$ ___ ___ ___
   3. There are 9 possible options for the digit $(1, 2, 3, 4, 5, 6, 7, 8, 9)$: $26 * 25 * 9$ ___ ___
   4. There are 10 possible option for the second and third digits $(0, 2, 3, 4, 5, 6, 7, 8, 9)$:  $26 * 25 * 9 * 10 * 10 = 585,000$