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Define Combination Formula: How Does It Work? Explain In Detail with Examples

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In the field of mathematics, the term combination is a collective selection of the items from a specific collection such that the order of selection which does not matter. As an example, if three fruits are given, such as an apple, an orange and a pear then there exists three different types of combination which can be extracted out from this set of three fruits. In a more formal way, a k-combination of a set S is a subset of k distinct elements of S. if the set consists of n elements, then the number of k-combinations is exactly equal to the binomial coefficient.

 

Which can be again written with the usage of factorials as (n!) / k!(n-k)! whenever k <= n, which is zero in case of k > n. The set of all the k-combinations of a set S is often denoted by.

Combination formula refers to the combination of n things taken k at the time without getting any repetition. To refer to such combinations in which repetition is allowed, the terms of k-selection, k-multi-set and k-combination with repetition is often used. If in the above mentioned example, it was possible to have two kinds of fruits then there would have the existence of 3 more 2-selections, among which one would have been with 2 apples and one with the oranges and the last one with two pears.

However, the set of all the three fruits was much smaller than the necessity to write a complete list of combinations, keeping in consideration with large sets which takes the form of practical. As another example, a hand of poker cards can be described as a 5-combination (k=5) of cards from a 52 card deck (n=52). Every single of the 5 cards in one hand are all different from each other, while the order of the cards present on the hand does not matter. There is an existence of 2,598,960 such combinations and the chance of drawing any one hand at random is 1/ 2,598,960.

The number of k-combinations from an already given set S of n elements is often denoted in the elementary form of combinatorics texts by C (n, k), or by another variation such as that of, or even C k n. This same number however occurs in many of the other existing mathematical contexts where it is denoted by  which specifically occurs in the form of a coefficient known as the binomial formula, hence it is also referred to as the binomial coefficient. One can define ( for all the natural numbers which exists denoted by the letter k at once by the relation,

From which it is clear that,

To have a look at these coefficients the k-combinations is counted from S, firstly it can be considered as a collection of n distinct variables Xs labelled by the elements s of S, and then be expanded as the product over all the elements of S.

 

It has 2n distinct terms which correspond to all the existing subsets of S, where each of the subsets give the value of the product of the variable corresponding denoted by Xs. Now, setting all the values included in the Xs are equal to the unlabeled variable X, so that the multiplication result equals to label free variable X, so that the product becomes (1+X)n, the term related to each k-combination from S becomes Xk , so the existing coefficient of the similar power related to the outcome equals up to the number of such k-combinations which are present.

Binomial coefficient may be calculated in a number of ways. To have the grip over all the expressions up to (1+X)n, one can implement the recursion relation which is as follows,

For the condition of 0 < k < n, which again follows the steps of (1+X)n = (1+X)n-1 (1+X), which leads to the construction of the Pascal’s triangle.

For the determination of an individual binomial coefficient, the formula having a more practical approach is,

The numerator gives the amount of k-permutations of n, i.e., of sequences related to the k distinct elements of S, while on the other hand the denominator provides with the number of such existing k-permutations that showcase the given k-combination when the order is entirely ignored.

When k exceeds its own value by n/2, the above mentioned formula consists of factors common both to the numerator as well as the denominator and again cancelling them out puts forward the relation,

For 0 <= k <=n. This provides a symmetry that is evident from the fact of the binomial formula and can also be explained with the help of terms of the k-combinations by taking into consideration the complement of such a combination, which is an (n-k) combination.

Finally, there is the existence of a formula which shows the symmetry in a direct manner, and has the property of being easy to remember:

 

Where, n! Denotes the factorial of the n. it is obtained from the previous mentioned formula by having a product of the denominator and the numerator by (n-k)!, so it is certainly inferior as the method of computation to that formula.

The last started formula can be understood in a direct manner, by considering the n! Permutations of all the stated elements of S. Each of such permutation puts forward a k-combination by selecting the first k elements. There exists many of the duplicate selections: any of the combined permutation among the first k elements and of the final (n-k) elements among them again produces the same combination. Hence, this explains the division in formula.

From the above mentioned formulas relations between the adjacent numbers which have been mentioned within the Pascal’s triangle in all the three existing directions is given below,

 

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