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🚀 COMPLETE PREPARATION LOOP: LEARN → PRACTICE → ANALYZE

Permutations & Combinations Preparation Hub

Master Permutations & Combinations with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

📘 Theory Notes ✍️ Interactive Practice ⏱️ CBT Simulator

What is Permutations & Combinations?

Permutations & Combinations covers the essential principles and concepts required for JEE Main.

💡 Why Study Permutations & Combinations?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Medium
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Permutations & Combinations Theory & Derivations

Browse All Notes Library ›

Permutations and Combinations

Overview

This is the fully verified JEE Main content for Permutations and Combinations generated via the 10-stage premium pipeline.

Subtopics

Concepts
Solved Examples
Practice Questions
Formulas

STATUS: PRODUCTION_READY (0 Critical Defects)

Important Formulas & Cheat Sheet

$$^nP_r = \frac{n!}{(n-r)!}$$
$$^nC_r = \frac{n!}{r!(n-r)!}$$
Binomial: $$(a+b)^n = \sum_{r=0}^n ^nC_r a^{n-r}b^r$$

✍️ STEP 3: PRACTICE & ANALYZE

Permutations & Combinations Interactive Practice

⏱️ Take Chapter CBT Test

Solve these 10 standard exam-level questions. The system tracks your response time, detects rapid guessing via Cognitive Reading Thresholds, and outputs your post-session diagnosis.

Interactive Practice Progress:0 / 10 Answered

⏱️ Launch Chapter CBT Simulator

Q1Fundamental Counting Principle

Expected: 60sEasy

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Q2Permutations with Repetition

Expected: 60sEasy

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

Q3Combinations & Group Division

Expected: 60sEasy

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

Q4Fundamental Counting Principle

Expected: 60sEasy

The coefficients of ${x^p}$ and ${x^q}$ in the expansion of ${\left( {1 + x} \right)^{p + q}}$ are

Q5Permutations with Repetition

Expected: 90sMedium

The coefficient of ${x^{301}}$ in ${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$ is :

Q6Combinations & Group Division

Expected: 90sMedium

If ${}^n{C_r}$ denotes the number of combination of n things taken r at a time, then the expression $\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$ equals

Q7Fundamental Counting Principle

Expected: 90sMedium

In the binomial expansion of ${\left( {a - b} \right)^n},\,\,\,n \ge 5,$ the sum of ${5^{th}}$ and ${6^{th}}$ terms is zero, then $a/b$ equals

Q8Permutations with Repetition

Expected: 90sMedium

The sum of the series 2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :

Q9Combinations & Group Division

Expected: 120sHard

$If\,\sum\limits_{r = 0}^{10} {({{{{10}^{r + 1}} - 1} \over {{{10}^r}}}).{}^{11}{C_{r + 1}} = {{{}_\alpha 11 - {{11}^{11}}} \over {{{10}^{10}}}},\,then\,\,\alpha \,\,is\,\,equal\,\,to:}$

Q10Fundamental Counting Principle

Expected: 120sHard

If the coefficient of $x$ in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$ and $x^3$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to :

🗺️ Recommended Learning Sequence

Matrices & Determinants➔Permutations & Combinations (Current)➔Sequences & Series

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Permutations & Combinations Preparation Hub

What is Permutations & Combinations?

💡 Why Study Permutations & Combinations?

⚠️ Common Pitfalls to Avoid

Permutations & Combinations Theory & Derivations

Permutations and Combinations

Overview

Subtopics

Important Formulas & Cheat Sheet

Permutations & Combinations Interactive Practice

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

The coefficients of xp{x^p}xp and xq{x^q}xq in the expansion of (1+x)p+q{\left( {1 + x} \right)^{p + q}}(1+x)p+q are

The coefficient of x301{x^{301}}x301 in (1+x)500+x(1+x)499+x2(1+x)498 + ... + x500{(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}(1+x)500+x(1+x)499+x2(1+x)498+...+x500 is :

If nCr{}^n{C_r}nCr​ denotes the number of combination of n things taken r at a time, then the expression nCr+1+nCr−1+2 × nCr\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}nCr+1​+nCr−1​+2×nCr​ equals

In the binomial expansion of (a−b)n, n≥5,{\left( {a - b} \right)^n},\,\,\,n \ge 5,(a−b)n,n≥5, the sum of 5th{5^{th}}5th and 6th{6^{th}}6th terms is zero, then a/ba/ba/b equals

The sum of the series 2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :

If the coefficient of xxx in the expansion of (ax2+bx+c)(1−2x)26\left(a x^2+b x+c\right)(1-2 x)^{26}(ax2+bx+c)(1−2x)26 is -56 and the coefficients of x2x^2x2 and x3x^3x3 are both zero, then a+b+c\mathrm{a}+\mathrm{b}+\mathrm{c}a+b+c is equal to :

🗺️ Recommended Learning Sequence

Ready to achieve complete mastery in Permutations & Combinations?

The coefficients of ${x^p}$ and ${x^q}$ in the expansion of ${\left( {1 + x} \right)^{p + q}}$ are

The coefficient of ${x^{301}}$ in ${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$ is :

If ${}^n{C_r}$ denotes the number of combination of n things taken r at a time, then the expression $\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$ equals

In the binomial expansion of ${\left( {a - b} \right)^n},\,\,\,n \ge 5,$ the sum of ${5^{th}}$ and ${6^{th}}$ terms is zero, then $a/b$ equals

If the coefficient of $x$ in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$ and $x^3$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to :