Permutations and CombinationsMCQMTP Oct 21Question 1713 of 251
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5C1+5C2+5C3+5C4+5C5\displaystyle ^5C_1 + ^5C_2 + ^5C_3 + ^5C_4 + ^5C_5 is equal to _______.

Options

A30
B31
C32
D35
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Correct Answer

✅ Option b — 31

All Options:

  • A30
  • B31
  • C32
  • D35

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Detailed Solution & Explanation

We want to find the value of:
S=5C1+5C2+5C3+5C4+5C5S = ^5C_1 + ^5C_2 + ^5C_3 + ^5C_4 + ^5C_5
Recall the binomial expansion sum of combinations:
nC0+nC1+nC2+⋯+nCn=2n^nC_0 + ^nC_1 + ^nC_2 + \dots + ^nC_n = 2^n
Subtracting nC0\displaystyle ^nC_0 from both sides:
nC1+nC2+⋯+nCn=2n−nC0^nC_1 + ^nC_2 + \dots + ^nC_n = 2^n - ^nC_0
Since nC0=1\displaystyle ^nC_0 = 1:
nC1+nC2+⋯+nCn=2n−1^nC_1 + ^nC_2 + \dots + ^nC_n = 2^n - 1
For n=5\displaystyle n = 5:
S=25−1=32−1=31S = 2^5 - 1 = 32 - 1 = 31
Let's verify by direct addition:
S=5+10+10+5+1=31S = 5 + 10 + 10 + 5 + 1 = 31
This is arithmetically correct and matches Option B (31).
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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