Fundamental Counting and Introduction to Factorial Notation

      Hi, my name is Nurielle Gregorio and I'll recap the two lessons we learned today. First of all, the morning portion of precal class, started with Mr. Piatek introducing us with a brief tour of our class website, which he worked on over the weekend. As you may all know, the website is a very useful tool that can potentially help everyone with any math problems, and even just as a study guide.

      Afterwards, most of the morning class was based on expanding our knowledge with The Fundamental Counting Principle. Once again, we opened our precal unit 1 handbook, titled as Permutations, Combinations, and Binomial Theorem. There, we were introduced with some basic rules of Fundamental Counting.

    As you can see, the image states the keywords we should look for when solving Fundamental Counting problems. In brief:

 Keywords:

"AND", "BOTH"
= multiplication

"OR", "EITHER", "OPPOSITE" = addition



 We continued to work on some more problems, and we encountered on a problem that is based on both of the principles of Fundamental Counting. 

   Example 5: Suppose that the executive of the Manitoba Association of Mathematics Teachers consists of three woman and two men. In how many ways can a president and a secretary be chosen if: 

 a.) The president is to be female and the     secretary male? 

 Step 1: Knowing that the problem stated "and", therefore, we should use multiplication. Also because they are asking for two different groups, female AND male.
Step 2: Find the number of females and male, which in this case, there are 3 females, and 2 males.
Therefore, it should be written as
3/president female x 2/secretary male= 6

b.) The president is to be male and the secretary female?

The answer should not be differ from a.) because your are still multiplying male with female. However, it is written differently, now that male is to be president, and female to be secretary:
2/president male x 3/secretary female=6

c.) The president and secretary are to be of the opposite sex?

    Step 1: Going back to the keywords/rules shown earlier, it is stated that "OPPOSITE"= ADD. So, 6+6= 12


Factorial Notation 

The second part of precal, which was in the afternoon, included our second lesson called
FACTORIAL NOTATION. 
The symbol for factorial is an exclamation mark (!)

 I found a video that helped me understand the concept of factorial notation more. However, the video included addition questions which we did not learn yet, but other than that, the video is very clear and helpful!

More complex factorial questions

Throughout the class, we worked on more complex questions.

For these questions, your must really understand that n!= n(n-1)(n-2)...(3)(2)(1), where n is the component of the positive integers.

Examples e. and d. are both very straight forward, as long as you take in mind the numbers you can cancel. So, if the numerator and the denominator have the same number, they can cancel out. 



Example e.) (s-2)!/ (s+1)!

Step 1: Knowing the equation n!= n(n-1)(n-2)... we can tell that (s-2) cannot end up to be (s=1) because -2-1=-3.
Step 2: We now try (s+1)!, which actually works. 1-1=0 which is (s), 0-1= -1 which is (s-1). Therefore, -1-1=-2 which is (s-2)
Step 3: Now that we know (s-2)/(s+1)(s)(s-1)(s-2), we end up with (s-2) as numerator, and another (s-2) in the denominator, which we know cancels out.
step 4: Now that they have canceled out each other, we have to take note that the numerator is not 0 but 1, so we are left with 1/(s+1)(s)(s-1).

The last question we worked on in  class will most likely be on the provincial exam! It is also very straight forward but includes factoring.

Exam Question

Example 6: Solve: n(n-5)!= 24(n-6) 

Step 1: Knowing (n-6) cannot be (n-5), we take -5-1= -6 which is (n-6).

Step 2: Now, we see similarity between both sides which is (n-6). Therefore we can divide both side with (n-6) to cancel out. We are then left with n(n-5)=24 

Step 3: You then just move over 24 to the other side, which leaves us with n(n-5)-24=0 

Step 4: FACTOR!
*NOTE* the final answer is n=8, not both n=8 AND n=-3 because in most cases, like this one, a neg number would be an extraneous solution/root.


  To conclude, I believe I've covered most of the concept and problems we encountered today for class. It seems that I wrote a whole unit! Nevertheless, I hope I was able to help, and provide a nice summary and step by step of  what we learned!:)

                                                                                                                   -Nurielle Gregorio