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  4. If the number of fourteen digit numbers which can be formed by using the digits 1,2,3,4,7,7, 7, ldots .7 (digit 7 is used 10 times) such that there are atleast two identical digits between distinct digits (distinct digits are 1,2,3 and 4 ) is k ⋅ 5 ! then k is

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Q. If the number of fourteen digit numbers which can be formed by using the digits $1,2,3,4,7,7$, $7, \ldots .7$ (digit 7 is used 10 times) such that there are atleast two identical digits between distinct digits (distinct digits are $1,2,3$ and 4 ) is $k \cdot 5$ ! then $k$ is

Permutations and Combinations

Solution:

$x_{1} \,\,1 x_{2} \,\,2 \,\, x_{3} 3 \,\,x_{4} 4 \,\,x_{5}$
where $x_{i}$ denotes no. of $7 's$
where $x_{2}, x_{3}, x_{4} \geq 2$
and $x_{1}, x_{5}>0$
Using beggar method, we get
${ }^{8} C_{4} \times 4 !=14 \times 5 !$