Combination:

What is Combination?

The arrangement of objects where the order of selection does not matter is called combination.

Example: The arrangement of three objects A, B, and C can be anything ABC or CBA or BCA or ACB or CAB or BAC. Here all the arrangements are equal but in permutation, it is not true.

Difference between permutation and combination: In permutation the "order" of the arrangement of the objects is necessary but in combination, the "order" of the arrangement of objects "does not matter".

Notation of combination:

If there are "n" number of distinct objects and we select "r" number of objects from them then the number of combinations is denoted as $$nC_r$$ or $$C(n,r)$$. $$nC_r = \frac{n!}{r! \times (n - r)!}$$ Where, n = Total number of objects.

r = Number of selected objects.

C = Combination.

Note: (i). $$nC_n = 1$$ (ii). $$nC_0 = 1$$ (iii). $$If \ nC_x = nC_y$$ then $$(x = y)$$ or $$x + y = n$$

Example(1): Find the value of $$4C_3$$?

Solution: $$nC_r = \frac{n!}{r! \times (n - r)!}$$ $$4C_3 = \frac{4!}{3! \times (4 - 3)!}$$ $$= 4$$

Example(2): If $$nC_5 = nC_{10}$$ then find the value of n?

Solution: We know that if $$nC_x = nC_y$$, then x + y = n

Hence $$nC_5 = nC_{10}$$ $$5 + 10 = n$$ $$n = 15$$