(C) The number consists of $2$ pairs of identical digits and $2$ distinct digits.Step $1$: Select the $2$ digits that will form the pairs from the $4$

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(C) The number consists of $2$ pairs of identical digits and $2$ distinct digits.Step $1$: Select the $2$ digits that will form the pairs from the $4$ available digits: $^4C_2 = 6$ ways.Step $2$: Select the $2$ remaining distinct digits from the remaining $2$ digits: $^2C_2 = 1$ way.Step $3$: Arrange these $6$ digits (where $2$ pairs are identical) in a row: $\frac{6!}{2! \times 2!} = \frac{720}{4} = 180$ ways.Total numbers = $6 \times 1 \times 180 = 1080$.

Class 11 Mathematics Permutation and Combination Definition of permutation, Number of permutations with or without repetition, Conditional permutations

Medium

How many $6$-digit numbers can be formed using the digits $1, 2, 3,$ and $4$ such that the number contains exactly two pairs of identical digits?

A
$480$
B
$540$
C
$1080$
D
None of these

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Permutation and Combination

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Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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How many $6$-digit numbers can be formed using the digits $1, 2, 3,$ and $4$ such that the number contains exactly two pairs of identical digits?

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