Hi everyone, it's Vienna! On Friday, we learned "Permutations with Case Restrictions". Now if the last lessons were quiet challenging, this topic might confuse you even more. In this lesson, there are more restrictions demonstrated.
For example: How many 3-digit numbers greater than 300 can you make using the digits 1,2,3,4,5 and 6? No digits are repeated. (note: I will be using this question throughout the explanation)
In order to solve the question you must understand when to use and what cases mean. What Case restriction means is that there are different possibilities that you may use that fits in the question.
In the question above, it is asking for a number/numbers that are greater than 300; Well from 1 to 6, it acts as if it's in the hundreds. Like 1 is equivalent to 100 (depending on the question given). In this case, the number that is greater than 300 can be 3. But that is not it, there is also 4,5,6 that is greater than 300.
Cases:
300 and above > 300
400 > 300
500 > 300
600 >300
First Step: For questions like this, it is highly common you use the dash method just because there will be plenty of restrictions. On the example, it instructs you to create 3-digit numbers, that means that N=3. Since there are 4 numbers that are greater than 300, that means you must create 4 columns of the dash method (note: column is preferably called "cases" in this topic.)
First case with 3 dashes _____ x _____ x _____
Second case _____ x _____ x _____
Third Case _____ x _____ x _____
Fourth Case _____ x _____ x _____
Second Step: Plug in the numbers that are greater in the first dash. The reasoning to having 1 for the dash is because there is only one number given to you. If there are 3 2's, then you put 2.
3-> 1 x x _____
4-> 1 x x
5-> 1 x x
6-> 1 x x
Third Step: In order to fill in the 2nd and 3rd dash, you need to know how many is left. It is indicated that "No repetitions are allowed". Since there are 6 options of numbers and we used one number already, there are five left. That means that the 5 is located in the 2nd dash while there will be 4 left in the third dash.
3-> 1 x 5 x 4
4-> 1 x 5 x 4
5-> 1 x 5 x 4
6-> 1 x 5 x 4
Fourth Step: You calculate the total for each dash and it will give you the answer of 20. But that is only for 1 Case. You have to add all of the four cases and the final result is 80.
3-> 1 x 5 x 4 = 20
4-> 1 x 5 x 4 = 20
+
5-> 1 x 5 x 4 = 20
6-> 1 x 5 x 4 = 20
80
In some questions such as this, you are able to do a case in one-shot. If there are for possibilities, that means that you can include that on the first dash. you would have four instead of one.
3,4,5,6-> 5 x 5 x 4 = 80
When I said "some" I meant these type of questions "How many 4-digit odd numbers can you make using the digits 1 to 7 if he numbers must be less than 6000. No digits are repeated." For this example, you must know how to contribute your clues into the question; You have n=4, you must indicate the odd numbers in a dash, and the you will have 5 cases because 1,2,3,4,5 are less than 6000. It may sound complicated but it will click once you see a solution.
1-> 1 x 5 x 4 x 3 <-3,5,7 = 60
2-> 1 x 5 x 4 x 3 <-1,3,5,7 = 80
3-> 1 x 5 x 4 x 3 <-1,5,7 = 60
4-> 1 x 5 x 4 x 3 <-1,3,5,7 = 80
5-> 1 x 5 x 4 x 3 <-1,3,7 = 60
6-> 1 x 5 x 4 x 4 <-1,3,5,7 = 80
3406-> 1 x 5 x 4 x 4 <-1,3,5,7 = 80
Do you see the pattern?. When you look at the first case, 1 is not included in the 4th dash because it is already being used and it is an odd number. When you look at the second dash, two is an even number so all the odd numbers are being shown on the 4th dash. As for the 5 and 4, that's how many numbers are left. Do you kind of get it?
Everyone deserves a break from all this lesson. For that, here is a little treat and enjoy! (And yes I got permission from Lloyd).
Do You Wanna Build A Snowman - Lloyd Umali
https://www.youtube.com/watch?v=b7ke-xEscQs
Vienna Jaime
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