M03-12

Question

How many odd integers between 100 and 1000, do not contain the digit "5"?

A. 180
B. 196
C. 286
D. 288
E. 324

Bunuel · Accepted Answer

Official Solution:

How many odd integers between 100 and 1000, do not contain the digit "5"?

A. 180
B. 196
C. 286
D. 288
E. 324

Let's examine the possible options for each digit in these numbers:
 
The hundreds digit has 8 options: 1, 2, 3, 4, 6, 7, 8, or 9 (excluding 0 and 5).
 
The tens digit has 9 options: 0, 1, 2, 3, 4, 6, 7, 8, or 9 (excluding 5).
 
The units digit has 4 odd options: 1, 3, 7, or 9 (excluding 0, 2, 4, 5, 6, and 8).
 
Consequently, the total number of odd integers between 100 and 1000 that do not contain the digit "5" is  8 * 9 * 4 = 32 * 9 = 288.

Answer: D

ChrisL1988 · Answer

I did an alternative approach, although I admit, that the solution is far more easier.

901  numbers
 450  odd numbers
 - 50  numbers (remaining odd numbers from 501 to 599)
 - 8*5  numbers (8x times (100-range, 200-range till 900-range; excluding 500-range) the remaining odd numbers from 51 to 59 (5 numbers))
 - 8*9  numbers (8x times (100-range, 200-range till 900-range; excluding 500-range) the remaining 9 odd numbers out of every 10-range; excluding 50-range)
 =288  numbers

ManSab · Answer

101,103,107, 109... total + 4
111,113,117,119.... total + 4
.......
151, 153....... don't count
.......
.......
191,193,197,199..... total +4
=====================
Total = 9*4 =36
Repeat same for numbers with 2 at hundred,3 at hundred,4 at hundred, 6 at hundred..... 9 at hundred position.

Therefore, total odd numbers that don't contain digit '5' are 36 * 8 = 288

Thanks

HunterJ · Answer

Last digit (units): 4 choices (1, 3, 7, 9 - cannot be 0, 2, 4, 5, 6, 8)

Why can't the last digits be anything even?

Bunuel · Answer

The question asks about the number of odd integers. An odd integer cannot have even units digit.

mafuz · Answer

Brilliant!The problem is just solved using counting method.

prashanthichennupati · Answer

I think this is a  high-quality  question and I agree with explanation.

JIAA · Answer

HI  Bunuel

Can you please provide link to Qs similar to this ??

Thanks!

Bunuel · Answer

Check  Constructing Numbers, Codes and Passwords Questions  topic from our  Special Questions Directory .

Hanzel101 · Answer

Bunuel chetan2u souvik101990 VeritasKarishma egmat

Hoping somebody would reply.

Though I agree with Bunuel's solution, I wanted to understand what is wrong in my approach.

_ _ _

In 1st dash, any number except for 0 & 5 ---- 8
In 2nd dash, any number except for 5 ---- 9
In 3rd dash, any number except for 5 ---- 9

Now we will get 8 * 9 * 9 = 648. Half of these numbers are even, other half odd. So, 648/2 = 324

It'll be great if somebody replies!

Bunuel · Answer

Not half of them will be odd. You excluded 5, so 4/9th will be odd. 648*4/9 = 288.

Bunuel · Answer

You are subtracting more odd numbers with 5, then there are. For example, number 515 is counted in the first group as well as in the third one.

Kritisood · Answer

Bunuel  your method is too good. I spent forever on this question analyzing every condition.

DixitNiket · Answer

Invested almost 21 minutes on this question to solve through number system. But didn't think through to use permutations and combinations.

carouselambra · Answer

yes exactly. I literally freaked out thinking about all the solutions only to realise that it was so so simple.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

assassinofkings98111 · Answer

Another way to do this question

We get 19 unique instances of 5 in every 100 numbers

Out of these 19 numbers, 5 are even namely x50, x52, x54, x56, x58. Therefore, you have a total of 19-5=14 odd numbers containing 5.

So for every 100 excluding 500-600, you get 14*8 = 112 odd numbers containing 5

Remember, there are 50 odd numbers between 500-599

So total number of odd numbers containing 5 is 112+50=162

Total number of odd integers between 100 and 1000 is 450

Subtract 450-162 = 288 - The final answer

Mahue · Answer

Hello,
I did it in a worst way (more time spended).

-Between 100 and 1000 there are 900 numbers.
- all 500´s (501,502...599) = 100
- Numbers containing a 5 : every 100 there are 10 (50,51,52...59) + 9 (105,115,125,135,145,165..) = 19*8 (not counting 500´s) = 152
- All odd numbers (5 every 10 without counting 50´s and 500´s) = 5*9*8 = 360
Total:
900-100-152-360 = 288

Emundtm · Answer

I think this is a  high-quality  question and I agree with explanation.

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