A combination is a grouping or selection of all or part of a number of things without reference to arrangement of the things selected. Selection of three members from a family of ten, number of cricket matches in a group having 10 teams, number of triangles formed by joining any three points out of 20 points in plane etc. all are example of combination. The symbol \({}^n{C_r}\) represents total number of selections taking r different objects out of n different objects.

\({}^n{C_r}\) =\(\cfrac{{n!}}{{(n - r)!(r!)}}\) 

=\(\cfrac{{{}^n{P_r}}}{{r!}}\)=\(\cfrac{{n(n - 1)(n - 2)...3.2.1}}{{r!}}\)

Suppose there are 5 students and a company wants to select any 3 of them. So the total number of ways = selection of 3 out of 5 = rejection of 2 out of 5. 

Number of ways = 5C3 = 5C2 or in general,

\({}^n{C_r}\) = \({}^n{C_{n - r}}\)

\(^{n + 1}{C_r}{ = ^n}{C_r}{ + ^n}{C_{r - 1}}\)

Suppose there are n + 1 students and we have to select r students out of these students. Now there are two cases:

Case 1: A particular student X is always selected, total number of ways of selecting remaining r – 1 students from the remaining n students =\(^n{C_{r - 1}}\)

Case 2: A particular student X is never selected, total number of ways of selecting r students from the remaining n students =\(^n{C_r}\). Hence total number of ways =\(^n{C_{r - 1}}\)+\(^n{C_r}\). 

Thus \(^{n + 1}{C_r} = {\;^n}{C_r}\,\, + \,{\,^n}{C_{r - 1}}\)

Example 01: There are 20 non collinear points in a plane. How many distinct triangles can be formed by joining any three points?

Example 02: In how many ways we can make two groups of 10 and 3 students out of a total 13 students.