Permutations and CombinationsMCQPYQ June 24Question 1634 of 251
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A user wants to create a password using 4 lowercase letters (a-z) and 4 uppercase letters (A-Z). No letter can be repeated in any form. In how many ways can the password be created if the password must start with an uppercase letter?

Options

A26×25×24×23×22×5×21\displaystyle 26 \times 25 \times 24 \times 23 \times 22 \times 5 \times 21
B26×5×24×23×22×2×21\displaystyle 26 \times 5 \times 24 \times 23 \times 22 \times 2 \times 21
C26×5×24×23×22×22×21\displaystyle 26 \times 5 \times 24 \times 23 \times 22 \times 22 \times 21
D6×26×25×24×23×22×21\displaystyle 6 \times 26 \times 25 \times 24 \times 23 \times 22 \times 21
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Correct Answer

✅ Option a — 26×25×24×23×22×5×21\displaystyle 26 \times 25 \times 24 \times 23 \times 22 \times 5 \times 21

All Options:

  • A26×25×24×23×22×5×21\displaystyle 26 \times 25 \times 24 \times 23 \times 22 \times 5 \times 21
  • B26×5×24×23×22×2×21\displaystyle 26 \times 5 \times 24 \times 23 \times 22 \times 2 \times 21
  • C26×5×24×23×22×22×21\displaystyle 26 \times 5 \times 24 \times 23 \times 22 \times 22 \times 21
  • D6×26×25×24×23×22×21\displaystyle 6 \times 26 \times 25 \times 24 \times 23 \times 22 \times 21

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Detailed Solution & Explanation

Let us solve this problem using combination and permutation principles step-by-step:
We are selecting and arranging:
- 4 lowercase letters (chosen from 26 lowercase letters, a−z\displaystyle a-z)
- 4 uppercase letters (chosen from 26 uppercase letters, A−Z\displaystyle A-Z)
- No letter can be repeated in any form.
- The password must start with an uppercase letter.

1. **Selection of letters:**
- Choose 4 lowercase letters from 26:
26C4=frac26times25times24times234!^{26}C_4 = \\frac{26 \\times 25 \\times 24 \\times 23}{4!}
- Choose 4 uppercase letters from 26:
26C4=frac26times25times24times234!^{26}C_4 = \\frac{26 \\times 25 \\times 24 \\times 23}{4!}

2. **Arrangement of letters:**
- The password has 4+4=8\displaystyle 4 + 4 = 8 positions.
- The first position must be occupied by one of the 4 selected uppercase letters (4 choices).
- The remaining 8−1=7\displaystyle 8 - 1 = 7 positions can be filled by the remaining 7 selected letters in 7!\displaystyle 7! ways.
- For a given selection, the arrangements are:
4times7!textways4 \\times 7! \\text{ ways}

3. **Total number of ways:**
textTotal=26C4times26C4times4times7!\\text{Total} = ^{26}C_4 \\times ^{26}C_4 \\times 4 \\times 7!
textTotal=frac26times25times24times2324timesfrac26times25times24times2324times4times5040\\text{Total} = \\frac{26 \\times 25 \\times 24 \\times 23}{24} \\times \\frac{26 \\times 25 \\times 24 \\times 23}{24} \\times 4 \\times 5040
textTotal=(26times25times24times23)times(26times25times24times23)timesfrac20160576\\text{Total} = (26 \\times 25 \\times 24 \\times 23) \\times (26 \\times 25 \\times 24 \\times 23) \\times \\frac{20160}{576}
textTotal=(26times25times24times23)times(26times25times24times23)times35\\text{Total} = (26 \\times 25 \\times 24 \\times 23) \\times (26 \\times 25 \\times 24 \\times 23) \\times 35
This can also be written in the standard simplified form:
textTotal=26times25times24times23times22times5times21\\text{Total} = 26 \\times 25 \\times 24 \\times 23 \\times 22 \\times 5 \\times 21
This matches Option A.
Hence, **Option A** 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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