GATE2019 EE: GA-6

Question

How many integers are there between $100$ and $1000$ all of whose digits are even?

• A. $60$
• B. $80$
• C. $100$
• D. $90$ 

Comments

$\text{Between}\: 100\: \text{and}\: 1000\: \text{means}:\: 100< \text{Numbers} < 1000$

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this question is tagged under arithmetic-series on GO pdf but this is clearly a counting problem

Answer 1

Let the 3 digit even number be ${\color{Red} \bigstar}\;{\color{Green} \bigstar}\;{\color{Blue} \bigstar}$

We can fill ${\color{Red} \bigstar}\;$ with $4$ values ($2,4,6,8$)

We can fill ${\color{Green} \bigstar}\;$ with $5$ values ($0,2,4,6,8$)

We can fill ${\color{Blue} \bigstar}\;$ with $5$ values ($0,2,4,6,8$).

So total $3$ digit even number possible = $4*5*5$ = $100$

$\therefore$ Option C $100$ is the correct answer.

Comments

$5\times5\times4 \neq 80$

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corrected...silly mistake.

Answer 2

From 100 to 199 there is no number exist whose all digits are even (0,2,4,6,8)

From 200 to 208,220 to 228,240 to 248,260 to 268,280 to 288   = 5+5+5+5+5= 25

Same as 400 to 488 have 25 total numbers

600 to 688 have 25 total numbers

800 to 888 have 25 total numbers

Hence integers between 100 and 1000 whose digits are even =25+25+25=100

answer C.