If $^{20}C_x = ^{20}C_{x+6}$, Then the value of $x$ is

The correct answer is Option b: 7. Given equation: $$^{20}C_x = ^{20}C_{x+6}$$ We use the general combinatorial property: If $^{n}C_{a} = ^{n}C_{b}$, then either: 1. $a = b$, or 2. $a + b = n$ Let us evaluate both cases: **Case 1:** $x = x + 6$ $$\implies 0 = 6$$ This is a contradiction, so Case 1 yields no solution. **Case 2:** $x + (x + 6) = 20$ $$\implies 2x + 6 = 20$$ $$\implies 2x = 14$$ $$\implies x = 7$$ Let us verify by substituting $x = 7$: $$\text{LHS} = ^{20}C_7$$ $$\text{RHS} = ^{20}C_{7+6} = ^{20}C_{13}$$ Since $^{20}C_{13} = ^{20}C_{20-13} = ^{20}C_7$, the LHS is indeed equal to the RHS. Hence, **Option B** is the correct answer.

Permutations and CombinationsMCQMTP June 2023 Series IQuestion 1737 of 251

All Questions

If $\displaystyle ^{20}C_x = ^{20}C_{x+6}$ , Then the value of $\displaystyle x$ is

Options

A10

C11

DNone of these

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Correct Answer

✅ Option b — 7

All Options:

A10
B7
C11
DNone of these

Detailed Solution & Explanation

Given equation:

^{20}C_x = ^{20}C_{x+6}

We use the general combinatorial property:
If

\displaystyle ^{n}C_{a} = ^{n}C_{b}

, then either:
1.

\displaystyle a = b

, or
2.

\displaystyle a + b = n

Let us evaluate both cases:

**Case 1:**

\displaystyle x = x + 6

\implies 0 = 6

This is a contradiction, so Case 1 yields no solution.

**Case 2:**

\displaystyle x + (x + 6) = 20

\implies 2x + 6 = 20

\implies 2x = 14

\implies x = 7

Let us verify by substituting

\displaystyle x = 7

\text{LHS} = ^{20}C_7

\text{RHS} = ^{20}C_{7+6} = ^{20}C_{13}

Since

\displaystyle ^{20}C_{13} = ^{20}C_{20-13} = ^{20}C_7

, the LHS is indeed equal to the RHS.

Hence, **Option B** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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If 20Cx=20Cx+6\displaystyle ^{20}C_x = ^{20}C_{x+6}20Cx​=20Cx+6​, Then the value of x\displaystyle xx is