Mathematics · Permutations and Combinations

JEE Main 2026 — 28 January, Morning Shift — Question 7

Let $S=\{1,2,3,4,5,6,7,8,9\}$ . Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Option A:
$29 x=5 y$
Option B:
$45 x=7 y$
Option C:
$21 x=4 y$
Correct
Option D:
$56 x=9 y$

Answer: C

Step-by-step solution

$\mathrm{S}=\{1,2,3, \ldots ., 9\}$

$\mathrm{x}={ }^{9} \mathrm{C}_{1} \cdot{ }^{8} \mathrm{C}_{7} \times \frac{9!}{2}=\frac{9 \times 8 \times 9!}{2}$

$\mathrm{y}={ }^{9} \mathrm{C}_{2} \cdot{ }^{7} \mathrm{C}_{5} \times \frac{9!}{2!\times 2!}=\frac{9 \times 8}{2} \times \frac{7 \times 6}{2} \times \frac{9!}{2!\times 2!}$

$\Rightarrow \frac{x}{y}=\frac{4}{21}$

$21 \mathrm{x}=4 \mathrm{y}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 28 January, Morning Shift
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Combinations