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Q.
The numbers greater than 1000000 that can be formed by using the digits $1,2,0,2,4,2,4$ are
Permutations and Combinations
Solution:
Since, $1000000$ is a $7$-digit number and the number of digits to be used is also $7$. Therefore, the numbers to be counted will be $7$ digit only. Also, the numbers have to be greater than $1000000$ , so they can begin either with $1,2$ or $4$.
The number of numbers beginning with $1=\frac{6 !}{3 ! 2 !}=\frac{4 \times 5 \times 6}{2}=60$ as when 1 is fixed at the extreme left position, the remaining digits to be rearranged will be $0 , 2,2,2,4,4$ in which there are $3, 2's$ and $2, 4's$.
Total numbers begining with $2$
$=\frac{6 !}{2 ! 2 !}=\frac{3 \times 4 \times 5 \times 6}{2}=180$
and total numbers begining with $4=\frac{6 !}{3 !}=4 \times 5 \times 6=120$
Therefore, the required number of numbers $=60+180+120=360$.
Alternate Method
The number of 7-digit arrangements, clearly, $\frac{7 !}{3 ! 2 !}=420$ But this will include those numbers also, which have 0 at the extreme left position. The number of such arrangements $\frac{6 !}{3 ! 2 !}$ (by fixing 0 at the extreme left position) $=60$
Therefore, the required number of numbers $=420-60=360$.