“Counting: JEE Problem Set” is available to Standard subscribers.
The counting problems JEE sets most: circular arrangements, distributing objects into groups, the gaps method, and counting by complements and cases, with a full timed paper.
3Lessons12Problems2Python Labs1Question Set1Mock Test
Circular arrangements: Round a table, everyone moving one seat gives the same seating, so n people sit in (n−1)! ways, not n!. Fix one person and fill the other seats from there. A necklace or a garland can also be turned over, so n different beads make 2(n−1)!
necklaces. Handle conditions as in a row: tie people together, or count the opposite.
Distributing objects: Sharing n identical objects among k different boxes is like placing n stars and k−1 bars in a row: (k−1n+k−1) ways, or (k−1n−1) when no box may be empty. The same count gives the whole-number solutions of x1+⋯+xk=n. Different objects go into k boxes in kn ways. Groups of equal size with no names need a division.
Counting by complement and cases: For "at least one", count everything and take away the cases with none. Split a hard count into cases that do not overlap and that cover everything, then add. For "no two together", place the others first and choose gaps. When sets overlap, add their sizes, take away the overlaps of pairs, and add back the overlap of all three. Derangements count orders with nothing in its own place.