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Every permutation, combination and probability question in a competitive paper turns on one question you ask before you write anything: does the order matter?
A cricket team of eleven chosen from fifteen players — order does not matter, the team is the same whoever is named first. The batting order of those eleven — order is everything. That single distinction separates nCr from nPr, and a student who asks it out loud before touching the calculator gets these questions right almost every time.
Probability then sits on top as one line: favourable outcomes divided by total outcomes. Nothing more. The counting you just learnt is what supplies both numbers.
SSC CGL and CHSL ask one to three questions from this cluster in every shift, Railway NTPC and Group D ask the dice-and-coin versions, and BPSC General Studies uses the simpler card and dice cases. Three formulas cover the lot.
1. In how many ways can a committee of 3 members be chosen from 8 people?
2. In how many different ways can the letters of the word LEVEL be arranged?
3. Two fair dice are thrown together. What is the probability that the sum of the numbers shown is 9?
4. One card is drawn at random from a well-shuffled pack of 52 cards. What is the probability that it is a face card?
5. In how many ways can 8 persons be seated around a round table?
6. Three fair coins are tossed together. What is the probability of getting at least one head?
7. Twelve people at a meeting shake hands with one another exactly once. How many handshakes take place?
8. How many diagonals does a polygon of 10 sides have?