L-16 | Permutations and Combinations | "At Least One" Problems | JEE Mains & Advanced Maths |

Hello Future IITians and NITians! WELCOME to Integron Maths.

In this 16th lecture on the Permutations and Combination series, we tackle a very important concept for JEE Mains and Advanced: how to solve combination problems with the condition of selecting "at least one" or "none" This is a fundamental topic that forms the basis for many complex problems in combinatorics and probability.

We will start by understanding the core formula for selecting at least one distinct item, ⁿC₁ + ⁿC₂ + ... + ⁿCₙ = 2ⁿ - 1, and then apply it to solve a wide variety of problems. This lecture is crucial for students preparing for JEE 2026, 2027, and beyond.

📚 Key Concepts & Problems Discussed in this Lecture:

The "At Least One" Principle: Understanding the formula for selecting one or more items from a set of distinct objects
Problem 1 (Friends Invitation): In how many ways can Mohan invite one or more of his 8 friends to dinner?
Problem 2 (Subsets): Finding the number of subsets of a set with 10 elements containing at least 3 elements
Problem 3 (Question Paper): A candidate has to select questions from two sections, with at least one question from each section being compulsory
Problem 4 (Book Selection): A tricky problem where a student selects at least one and at most 'n' books from a collection of (2n+1) books
Problem 5 (Mixed Collections): Forming collections from books of different subjects (Physics, Chemistry, Maths) with various conditions
Problem 6 (Colored Balls): Creating groups from different colored balls where at least one of certain colors must be included
Problem 7 (Team Selection): Forming a team of boys and girls with conditions like "no girls" or "at least one boy and one girl"
Problem 8 (AIEEE 2006 PYQ): A previous year's question on voting in an election

Make sure you have your pen and paper ready to solve these problems along with me!

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Timestamps:
00:00 - Introduction
01:45 - Concept Overview "At Least One" & "None" Combinations
03:18 - Deriving the Formula: ⁿC₁ + ⁿC₂ + ... + ⁿCₙ = 2ⁿ - 1
12:54 - Problem 1: Inviting Friends
14:46 - Problem 2: Subsets with a Minimum Number of Elements
25:08 - Problem 3: Compulsory Selections in a Question Paper
35:19 - Problem 4: Finding 'n' from a Collection of (2n+1) Books
40:15 - Problem 5: Collections of Books from Different Subjects
46:48 - Problem 6: Selecting Different Colored Balls
50:14 - Problem 7: Team Selection of Boys and Girls
57:33 - Problem 8: AIEEE 2006 Previous Year Question on Voting
1:02:30 - Summary and What's Next



Selection of at least one distinct object

Combination formula and proofs for JEE

Solving complex PnC problems for JEE Advanced

Tricks for committee formation and team selection questions

Analysis of Previous Year Questions (PYQ) from AIEEE (JEE Main)

Fundamentals of grouping and distribution in combinatorics

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