Permutation, Combination, and Probability

Probability is the chance of an event occurring out of the total results considered. The probability in mathematics can be divided into three forms depending on their applicability, theoretical, practical, and axiomatic. The probable outcomes are calculated with or without the importance of the sequential orders, and this divides the mathematical problems of probability into two primary categories, permutation, and combination.

Probability

Probability is divided into two types of mathematics, permutation, and combination. It is usually expressed through fraction or decimal numbers. When the fraction is used, it is also simplified. 

Example

To calculate the probability the numerical value of the favorable results is divided by the total number of results. If I have gone to a shop and have picked 4 roses, 3 dahlias, and 2 orchids and have kept them in a closed basket, what is the probability of randomly picking up a rose? 

Here probability is used for calculating the possible result. The favorable result will be the number of roses, which is 4, and the total number of results will be the total number of flowers, which is 4 + 3 + 2 = 9. So, the probability of putting my hand in the basket and randomly picking up a rose is 4 / 9. 

Difference between Permutation and Combination 

When the sequence of the digits has to be considered, then it is a combination. When the sequence of the digits can be ignored and it is of no consequence, then it is a permutation. 

Permutation

To work out the solutions of probability mathematics with the use of permutation, first, the number of the probability of the event is considered, and then it is multiplied with the same number X times. Here X denotes the numerical value of the events in the sequential order.

Permutation can occur with or without repetition. The first scenario is where there might be the repetition of the outcomes and in the second one, that possibility is absent, because once an outcome occurs, it is eliminated from the next set of probabilities. 

Example

I have 4 different boxes and I have to calculate the probabilities of arranging them on a rack. We have to use the permutation formula here for calculating the probability. The number of probabilities will be 4 * 3 * 2 * 1 = 24. Here this is an example of permutation without repeat. To solve these problems easily, factorial is used. It is the multiplication of all the numbers from 1 to the taken number. Here n denotes that number. For example, the factorial of 5 is 5 ! = 5 * 4 * 3 * 2 * 1= 120.

Now we shall take the example of one with repetition. I have a 3 digit pin number as a password for my account. The probability of the pin number will be 10 * 10 * 10 = 1000. 

Sometimes permutation is used with partial repetition. Here the calculation for probability is done differently. Here the factorial of n is divided by the factorial of r, n denotes the total number of possibilities and r denotes the volume of the permutation. 

So, the formula will be n ! / r!

If we have 5 boxes and we can arrange them in 3 possible ways then the calculation for probability will be 5! / 3! = 5 * 4 * 3 * 2 * 1 / 3 * 2 * 1 = 20.

Combination 

To work out the solutions of probability mathematics with the use of the combination, factorial is used. The formula is n C r = n ! / r ! * ( n – r )! Here n denotes the total numerical value and r denotes the number of probabilities we are considering. C is the symbol for combination. The only constant thing is that n should always be greater than r.

Example

There are five new books in the market. I have to buy three of them. How many combinations could be there? 

The solution is 5  C3 = 5 ! / 3 ! * ( 5 – 3 ) ! = ( 5 * 4 * 3 * 2 * 1 / 3 * 2 * 1 )  * ( 2 * 1 )  =  20 * 2  = 40.

So I shall have 40 combinations for buying 3 books out of 5.

Conclusion

The use of probability is a significant part of quantitative aptitude. It is useful in real-life scenarios and can be beneficial in scientific experiments and in laboratories. However, the use of permutations and combinations for solving mathematical problems takes practice. But they are easy to learn and muster.