The number of four-digit natural numbers whic | vedclass

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Question

(A) To form a four-digit number with exactly two distinct digits,we consider two cases:Case $1$: The digit $0$ is not included.We choose $2$ digits ou

Vedclass · Accepted Answer

$567$

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(A) To form a four-digit number with exactly two distinct digits,we consider two cases:Case $1$: The digit $0$ is not included.We choose $2$ digits out of the $9$ non-zero digits $(1-9)$ in $^9C_2$ ways.For each selection,we can form $2^4 - 2$ numbers (total arrangements $2^4$ minus the cases where all digits are the same).Number of ways $= ^9C_2 	imes (2^4 - 2) = 36 	imes 14 = 504$.Case $2$: The digit $0$ is included.We choose $1$ non-zero digit out of $9$ in $^9C_1$ ways.The number must start with the non-zero digit. The remaining $3$ positions can be filled by the chosen non-zero digit and $0$ such that at least one $0$ is present.This is equivalent to $2^3 - 1 = 7$ ways.Number of ways $= ^9C_1 	imes 7 = 9 	imes 7 = 63$.Total numbers $= 504 + 63 = 567$.

The number of four-digit natural numbers which contain exactly two distinct digits is:

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