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

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, [2026]

Smart Achievers · Accepted Answer

(2)

$S = {1, 2, 3, \dots, 9}$

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

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

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

$\Rightarrow 21 x = 4 y$

Solution

1 $45 x = 7 y$

2 $21 x = 4 y$

3 $29 x = 5 y$

4 $56 x = 9 y$

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Solution

1 45x=7y

2 21x=4y

3 29x=5y

4 56x=9y

Ans. (2) S={1,2,3,…,9} x=C19·C78×9!2=9×8×9!2 y=C29·C57×9!2! 2!=9×82×7×62×9!2! 2! ⇒xy=421 ⇒21x=4y

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1 $45 x = 7 y$

2 $21 x = 4 y$

3 $29 x = 5 y$

4 $56 x = 9 y$