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Q. Number of $6$ digit numbers that can be made with the digits $1,2,3$ and $4$ and having exactly two pairs of digits is

Permutations and Combinations

Solution:

The number will have $2$ pairs and $2$ different digits.
The number of selections $={ }^{4} C _{2} \times{ }^{2} C _{2}$,
and for each selection, number of arrangements
$=\frac{6 !}{2 ! 2 !}$.
Thus, the required number
$={ }^{4} C _{2} \times{ }^{2} C _{2} \times \frac{6 !}{2 ! 2 !}=1080$