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Combination

 

In mathematics, the term combination means selection. It is used to find ways in which we can select objects in n number of ways. For example, we often want to choose the things where we do not care about the order or sequence of the selected objects. For example, if we decide on chocolates from a store irrespective of their sequence, we choose the combination.

Combination formula

In the combination formula, we choose n number of things in k ways. Therefore, it is known as the n choose k formula. It is denoted by n Ck, n Ck, or even n Ck. The combination is also known as a binomial coefficient. It is used to select n things in k ways without any specific order or sequence. The combination formula is:

image

where, n = total number of objects

k = number of objects that we want to choose

Combination formula derivation

The combination formula is derived from the fundamental principle of counting. We know things can be arranged in multiple ways. Therefore, choosing an object randomly can be done in numerous ways. The permutation is a special case of combination. In permutation, the sequence of things also matters. Therefore,

Permutation = selection AND arrangement

If k objects are selected from n number of things where order also matters, then from the fundamental principle of counting, we have,

P (n, k) = C (n, k) k!

We know that the permutation is equal to

n! / (n – k)!. Putting in the above formula we have,

n! (n – k)! = C (n, k) k!

image

Important results

image1

Example: Find the number of ways in which we can form a team of 6 cricketers out of 18 cricketers.

Solution:

Here, the total number of members is n = 18.

The number of members to be selected to form a team is k = 6.

This can be done in C (n, k) ways, so we will use the combination formula.

C (n, k) = n! / [(n-k)! k!]

C (18, 6) = 18! / [(18 - 6)! 6!]

C (18, 6) = (18 × 17 × 16 × 15 x 14 x 13 x 12!) / (12! 6!)

C (18, 6) = (18 × 17 × 16 × 15 x 14 x 13) / (6 x 5 x 4 x 3 × 2 × 1)

C (18, 6) = 720.

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