Permutation and CombinationHard
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 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,
Options
A.$29x = 5y$
B.$45x = 7y$
C.$21x = 4y$
D.$56x = 9y$
Solution
$S = \{ 1,2,3,\ldots.,9\}$
$$\begin{matrix} & x = \ ^{9}C_{1} \cdot \ ^{8}C_{7} \times \frac{9!}{2} = \frac{9 \times 8 \times 9!}{2} \\ & y = \ ^{9}C_{2} \cdot \ ^{7}C_{5} \times \frac{9!}{2! \times 2!} = \frac{9 \times 8}{2} \times \frac{7 \times 6}{2} \times \frac{9!}{2! \times 2!} \\ & \ \Rightarrow \frac{x}{y} = \frac{4}{21} \\ & 21x = 4y \end{matrix}$$
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