Derivations of formulas and their connections

Unequal equal-combination indices are complementary, so (r+r+3=15). In exams set the sum of lower indices equal to the upper index.

The complementary condition gives (2r+r+3=18), so (r=5). In exams consider two cases for equality: same index or complementary index.

Different indices with equal values are complementary. In exams add the indices and match them with (n).

Combination gives only the group and (r!) adds arrangements of that group. In exams treat permutation as selection plus arrangement.

To get unordered count from ordered count, (r!) orders are removed. In exams divide when order is ignored.

Subtract selections of (0), (1), and (2) from all subsets. In exams handle at least by total minus small unwanted cases.

At most (3) includes selections of (0), (1), (2), and (3). In exams add all allowed cases when there is an upper limit.

In stars and bars, (18) stars and (3) bars are arranged. In exams use (^{n+r-1}C_{r-1}) for non-negative solutions.

Giving (3) first to each variable leaves (12), then count non-negative solutions. In exams shift the minimum condition.

For identical pens and distinct students, use (9) stars and (3) bars. In exams remember stars and bars for zero-allowed distribution.

After giving (1) ball to each box, (6) balls remain. In exams allot the minimum first in positive distribution.

First choose the empty person, then distribute positively among the remaining (3) persons. In exams use choose empty plus positive stars and bars for exactly empty conditions.

Each distinct object has (r) independent choices. In exams use the power rule for distinct objects and unrestricted boxes.

Each prize has (10) independent choices. In exams use the power formula when distinct prizes and repeated recipients are allowed.

Prizes are distinct and recipients cannot repeat, so it becomes an ordered assignment. In exams treat distinct prizes without repetition as permutation.

In identical prize distribution, use (6) stars and (9) bars. In exams apply stars and bars for identical items and distinct receivers.

The three posts are different, so the order of selection is meaningful. In exams use permutation for different posts.

A committee without posts counts only the group. In exams use combination when there are no posts.

Any (3) points form a triangle and order is not important. In exams use combination in geometry selection.

Subtract collinear (3)-point selections from total (3)-point selections. In exams subtract invalid selections.

A selection of four vertices determines a quadrilateral. In exams use combinations of vertices when shapes are formed from polygons.

A handshake of (A) with (B) and (B) with (A) is the same. In exams treat mutual relations as combinations.

A match is determined by an unordered pair of two teams. In exams use (^{n}C_2) for one-to-one pair events.

Treating the (3) special books as one block gives (5) objects to arrange. In exams also count (3!) arrangements inside the block.

Subtract the arrangements where (A) and (B) are together as a block from total arrangements. In exams use complement for not together.