Permutations & Combinations
9-Digit Numbers with Repeated Digits — Ratio x:y
nta_pyq_2026_jan
Grade 11
Question:
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 $S$ 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 $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then
$56x=9y$
$29x=5y$
$45x=7y$
$21x=4y$
Step-by-Step Solution
Key Concept: $x$: choose repeated digit (9 ways), absent digit (8 ways), arrange $\tfrac{9!}{2!}$ ways. $x=9\times8\times\tfrac{9!}{2}=72\times181440$. $y$: choose 2 repeated digits ($\binom{9}{2}=36$), choose 2 absent digits ($\binom{7}{2}=21$), arrange $\tfrac{9!}{2!2!}$ ways.
$21x=4y$.
Correct Answer: 4