Combinatorics

Combinatorics is a branch of mathematics dealing with combinations of objects in specific sets under certain constraints. It includes counting, arrangement, and combination of elements.

A combination is a selection of items from a larger pool where the order does not matter. It is denoted as $$_nC_r$$ , where $$n$$ is the total number of items and $$r$$ is the number of items to choose.

A permutation is an arrangement of items where the order matters. It is denoted as $$_nP_r$$ , where $$n$$ is the total number of items and $$r$$ is the number of items to arrange.

20 (calculated using combinations: $$_6C_3 = \frac{6!}{3!(6-3)!} = 20$$ ).

The formula for combinations is $$_nC_r = \frac{n!}{r!(n-r)!}$$ .

n (This represents the number of ways to choose all but one item from n items.)

420 (This represents the number of ways to choose 3 items from 7, considering 2 identical items.)

Combination (This is a selection problem where the order of selection does not matter.)

Permutations consider the order of selection, while combinations do not.

720 (This is a permutation problem since the order of selection matters: $$10P_3 = 10!/(10-3)! = 720$$ ).