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  • shahkb4
    3 June, 2026 at 8:21 pm #1101

    The number of numbers greater than 5000 , less than 9000 and divisible by 3 , that can be formed using the digits $0,1,2,5,9$, if the repetition of the digits is allowed, is $\_\_\_\_$ [JEE MAIN 24 Jan 2026 S I]

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    shahkb4
    3 June, 2026 at 8:21 pm #1102

    Solution

    5 x y z

    $\quad \mathrm{x}, \mathrm{y}, \mathrm{z} \in\{0,1,2,5,9\}$

    \[ \frac{5+x+y+z}{3}=1+\frac{x+y+z+2}{3} \in I \]
    $\mathrm{x}+\mathrm{y}+\mathrm{z}$ $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ ways
    1 $0,0,1$ $3!/ 2!=3$
    4 $0,2,2$ $3!/ 2!=3$
    4 $1,1,2$ $3!/ 2!=3$
    7 $0,2,5$ $3!=6$
    7 $1,1,5$ $3!/ 2!=3$
    10 $0,1,9$ $3!=6$
    10 $0,5,5$ $3!/ 2!=3$
    13 $9,2,2$ $3!/ 2!=3$
    16 $2,5,9$ $3!=6$
    19 $9,9,1$ $3!/ 2!=3$
    19 $5,5,9$ $3!/ 2!=3$

    Total possible ways of $\mathrm{x}, \mathrm{y}, \mathrm{z}=42$ Answer(42)

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