How To Solve Permutation And Combination

There are stages in life and so are there stages in our learning phase as individuals. As a math student, understanding this simple and basic fact completely change the way I saw mathematical questions and most especially permutation and combination. 

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At first, solving it was a nightmare as the questions always look rather too confusing. But when I was able to understand the basic concept that surrounds it, solving permutation and combination questions became a thing I took delight in.  

Having seen issues from both sides, I understand the plights of students when it comes to solving permutation and combination questions and how they can be explained and understood easier. This has prompted me to share the knowledge I have gained so far with other students who are having a hard time understanding permutation and combination.

What is Permutation and how can I solve it? 

Permutation is a mathematical means of calculating how often a particular character reoccur in other or the possibilities of a number occurring in an order. Let’s take, for example, we have three cars, green(G), White(W), and blue(B) in first, second and third place respectively after a race. These same cars can be re-arranged in 6 different order which are B W G, B G W, W B G, W G B, G B W, and G W B. 

To solve more complex situations using this same formula,  factorial, “!” was introduced. Here, I was able to understand that I can use factorial to arrive at the same answer. Using the example above 3 cars=3! which is equivalent to 3 x 2 x 1=6 

Taking a step further, I also discover that permutation can also be used to solve other related questions. A good example to consider is when we have 5 cars which are in a race and we are asked to find the number of ways the first three can finish. Using permutation, it becomes 5P3. 

To solve for this, multiplication can’t be done directly as did earlier. Instead, I will only consider for the order that counts which is 5, 4, and 3. From this, 5P3= 5 x 4 x 3=60. Which means the first three cars can finish in 60 number of ways. 

Permutation can also be extended to when we have repeated elements. To help explain this, I will go over an example. Let’s say am asked to find the permutation of the word “KNEE” which has three letters with E being repeated twice. To solve for this, the formula becomes \frac{4P4}{2!}= \frac{4\times3\times2\times1}{2\times1}=12

What is a combination and how can I solve it? 

Unlike permutation, a fixed formula is been used to calculate combination and number of an order doesn’t count here. This formula is nCr\frac{n!}{r!(n-r)!} and to solve a question of 5C3, the equation changes to \frac{5!}{3!\,2!}. Using the same method as permutation, the final answer becomes 10. 

Just like every other maths calculations, understanding help build the foundation on which complex principles are built. And with my basic understanding of permutation and combination, I was able to do so.