Question

I. The number of all ten digited numbers that can be formed with all the distinct digits and which are divisible by $4$ is $15 \times 8 !$.
II. The number of positive integers that can be formed by using the digits $0,1,2,3,4,5$ without any repetition is $1630 $.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Both I and II are false

Answer 1

Answer: (B)

I. We have,
$1,2,3,4,5,6,7,8,9$
The number formed can be divisible by $4$ when last two digits has to be divisible by $ 4$ .
There are $16$ last two digits numbers that are divisible by $4$ without using zero.
So, total number of such $10$ digits numbers $=7 \times 7 ! \times 16$
Again, there are $6$ last two digits numbers that are divisible by $4$ with using zero.
So, total number of such $10$ digit numbers $=8 ! \times 6$
$\therefore $ Required numbers
$= 7 \times 7 ! \times 16+8 ! \times 6 $
$= 7 \times 8 ! \times 2+8 ! \times 6 $
$= 8 !(14+6)=20 \times 8 !$
So, statements I is wrong.
II. Six digit numbers $=5 \times 5 \times 4 \times 3 \times 2 \times 1=600$
Five digit numbers $=5 \times 5 \times 4 \times 3 \times 2=600$
Four digit numbers $=5 \times 5 \times 4 \times 3=300$
Three digit numbers $=5 \times 5 \times 4=100$
Two digit numbers $=5 \times 5=25$
One digit number $=5=5$
$\therefore $ Required number $ =600+600+300+100+25+5=1630$