Question

How many even numbers greater than $300$ can be formed with the digits $1,2,3,4,5$ if repetition of digits in a number is not allowed

• A. $110$
• B. $109$
• C. $111$
• D. $112$

Answer 1

The correct option is C $111$

No$:$ can be $3,4$ or $5$ digit$.$

$5$ digit even no$:$'s $:$

last digit must be either $2$ or $4,$ so $2$ choice$.$

First $4$ digit even no$:$ possible$=4!\ast 2=48$

$4$ digit even no$:$'s

last digit again must be either $2$ or $4.$ so $2$ choice i.e, $p_4^3\ast 2=48.$

$3$ digit even no$:$'s

last digit again must be either $2$ or $4.$ So $2$ choice i.e, $p_4^2=24.$

But out of these $24,3$-digit no$:$ some will be less than $300,$ the ones starting with $1$ or $2.$

These are $9$ such no$:$'s $(1\times 2$ and $3$ options for the $2^{nd}$ digit $x,$ so $3+(1\times 4)$ and $3$ options for $x,$ so $(3+2\times 4)$ and $3$ options for $x.$

$\therefore$ $24-9=15$

$\therefore$ Total $=48+48+15=111.$