Permutations

    Hi, my name is Hanz and this is a brief summary of today's lesson. Basically, Mr. Piatek started off the class answering a couple of questions from the Fundamental Counting Principle worksheet. And he also gave out possible solutions to yesterday's quiz.

    Eventually, he started the lesson which centered on a new topic called Permutations. According to our unit 1 handbook, a permutation of a set of distinct objects is an arrangement of objects, without repetition, into specific order. To permute a set of objects means to rearrange them. Give below is the formula for permutations:

                                                                

                                                     

 Further into the topic, Mr. Piatek went over the first example given in the book and analyzed simple rules of permutations. 

    When arranging items in a specific order, you can either use the permutation formula or the dash method.

 The dash method can be used for all situations but the permutation formula can only work on questions that don't allow for repetition and/or has no restrictions. 

   Repetitions means you're allowed to choose the same item more than one. While restrictions are when you are asked to place an item in a specific location. 

If the question does allow repetition and/or restrictions, you must use the dash method.

    Example 1: How many three letter words composed from the 26 letters in the alphabet are possible if...

    a) no repetitions are allowed?

                             

     The first step to solve the problem is to find the amount of items to that can be used as substitutes for r and n. Now take note that n ≥ r, meaning n could be any number greater than r. In this case, n = 26 and r = 3. Next step is to plug in the numbers in the formula and solve the question by simplifying.

     b) repetitions are allowed?

                             

      Now, as I said earlier, if the question does allow repetition and/or restrictions you must use the dash method. Therefore, you cannot use the formula. Assume no repetitions are allowed unless otherwise stated.

Part two

    In the afternoon, we had our second class of the day, and Mr. Piatek carried on about permutations and went through the second part of the lesson. We went over a few of the examples in the handbook like we did earlier and got introduced to a new formula.

                                                       

     This formula shows that the total number of permutations (as if there were no identical objects) must be divided by the number (in factorial form) of repeated, identical or like objects. 

     So we pretty much spent our whole class time going over the examples on the handbook similar to the ones above. But we also did encounter some that involved a little bit of thinking.

    Such as this one:

                             

     Letter c of example 2 in the handbook seemed a bit like all the other problems we discussed so far, not until someone pointed out an error in the answer. The question stated that the vowels are to be put in the middle but did not specify which position the vowels are to be placed.

   As you can see at the bottom of the picture, a and e are placed beside each other with number 1 placed above each letter. This indicates the total amount of vowels in the word MAPLES, which is two. But since it isn't stated in the question where the vowels are supposed to be placed, a and e are replaced with the letter v, to specify the restriction. 

   And also because of this, without any specification of any position for both of the vowels, and because the total number of vowels is two, the number two is placed above the first v while the number 1 is placed above the second v. Since the first vowel took one of the two positions in the middle, where both vowels should be placed, the second vowel takes the second place. And the rest of the spaces are filled with whatever number of letters are left.

    So all in all, that's it. I tried my hardest and best to sound smart and cover most of the stuff we learned today in class. I hope i did well in both and good luck to all of you. 

:)