Q.Out of 120 students, two sections of 50 and 70 are formed. If you and your friend are among the 120 students, find which is more likely to happen that you both enter the same section or different sections.

Combinations Probability: Counting When Order Doesn't Matter

Imagine you are picking a team of 3 students from a class of 10. You do not care who is chosen first, second, or third — you only care which 3 students end up on the team. That is a combination: a selection where order does not matter.

If you now ask for the chance that one particular set of 3 students (say your three best friends) is the one chosen, you are doing combinations probability: probability where the favourable and total outcomes are both counted using combinations.

The Core Intuition

When every possible selection is equally likely (like drawing names from a hat), the probability of an event is the familiar ratio:

This is the same "favourable over total" idea from basic probability — the only new part is that we count combinations, not arrangements, because order is irrelevant.

The Precise Statement

Let $n$ be the total number of distinct objects and let $r$ be how many you choose (without replacement). The number of ways to choose $r$ objects from $n$ is:

read as "$n$ choose $r$". If a selection of $r$ objects is made at random and every combination is equally likely, then the probability of an event $E$ is:

A Worked Example

Problem: A bag has 5 red marbles and 3 blue marbles. You draw 3 marbles at random (without looking). What is the probability of getting exactly 2 red marbles?

Step 1 — Total outcomes. Choosing 3 marbles from 8:

Step 2 — Favourable outcomes. You need exactly 2 red (from 5) and 1 blue (from 3):

Step 3 — Probability.

Combinations vs. Permutations

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