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Study Stuff for MCA Examinations - NIMCET
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Previous Year Question (PYQs)
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,
Solution
$ S = {1, 2, 3, \ldots, 9} $
$ x = {^9C_1} \cdot {^9C_2} \cdot \frac{9!}{2} = \frac{9 \times 8 \times 9!}{2} $
$ y = {^9C_2} \cdot {^7C_2} \cdot \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} $
$ \Rightarrow 21x = 4y $
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