How many of the integers from 100 to 999, inclusive, do NOT contain ei

Question

How many of the integers from 100 to 999, inclusive, do NOT contain either the digit 2 or the digit 5?

A. 343
B. 448
C. 512
D. 648
E. 700

gmatophobia · Accepted Answer

_ _ _

The hundreds digit can be filled by 1, 3, 4, 6, 7, 8, or 9. Hence, the hundreds digits can be filled in 7 ways

The tens digit can be filled by 0, 1, 3, 4, 6, 7, 8, or 9. Hence, the tens digits can be filled in 8 ways

The unit digit can be filled by 0, 1, 3, 4, 6, 7, 8, or 9. Hence, the unit digits can be filled in 8 ways

Total number of combinations = 8 * 8 * 7 = 448

Option B

MartyMurray · Answer

How many of the integers from 100 to 999, inclusive, do NOT contain either the digit 2 or the digit 5?

Many GMAT Quant questions can be solved in multiple ways. For instance, in the case of this question, if you didn't know how to use the approach  gmatophobia  used, you could use brute force.

Starting number of integers: 900

Take away all the 200 and 500s: 700

Take away all the remaining 20s and 50s: 700 - 7(10 + 10) = 560

Take away a 2 and a 5 for each of the remaining 10s: 560 - 7(8 × 2) = 560 - 112 = 448

A. 343
B. 448
C. 512
D. 648
E. 700

Correct answer:  B

Account9855 · Answer

Why doesnt this approach work?

Use either equation (for non mutually exclusive events)

=> Total # of integers = # of integers that dont have 2 + # of integers that dont have 5 - # of integers that dont have 2 and 5

Since the problem is asking for # of integers that dont have EITHER a 2 or 5, wont this also include integers that dont have a 2 but have a 5 and integers that dont have a 5 but have a 2?

What logical mistake am I making?

I appreciate any response!!!

Bunuel · Answer

The phrase "do NOT contain either the digit 2 or the digit 5" means we are looking for numbers that do not contain either of these digits.

Ggt1234 · Answer

Is this the same as asking for numbers that contain neither 2 nor 5?

Krunaal · Answer

Yes, numbers that  do NOT contain either the digit 2 or the digit 5  is the same as numbers that  contain neither 2 nor 5

ManifestDreamMBA · Answer

Hundreds digit option = 10-3=7 (cannot be 0 or 2 or 5)
Tens digit option = 10-2=8 (cannot be 2 or 5)
Units digit option = 10-2=8 (cannot be 2 or 5)
No constraint on repetition, so the number of integers = 7*8*8 = 448

KarishmaB · Answer

Responding to a pm: The easiest method here is to use combinations only as discussed above too. I have discussed this method here: https://youtu.be/_C-kTA44OxE We need all 3 digit numbers without 2 and 5. __ ___ ___ The first digit can be chosen in 7 ways (1 to 9 except 2 and 5) The second digit can be chosen in 8 ways (0 to 9 except 2 and 5) The third digit can be chosen in 8 ways (0 to 9 except 2 and 5) Total = 7 * 8 * 8 The answer must end with 8 and should be less than 8*8*10 = 640. Hence it must be 448 Answer (B)

Nsp10 · Answer

Bunuel 
pls help me with the alternate approach
total - digits that have 2 or 5 in them

Bunuel · Answer

Here is that approach: https://gmatclub.com/forum/how-many-of- ... l#p3395130 Hope it helps.

SudsMeister · Answer

Total = numbers between 100 and 999 inclusive = 900.

200-299 and 500-599 will each have 100 numbers which contain 2 or 5.

For remaining each set of 100 count numbers having 2 or 5.
Let's look at 100 - 199, for instance.

Start with 2s:
102, 112, 122, 132, 142, 152, 162, 172, 182, 192 = 10
120 - 129 has 10 numbers but we have already counted 122 above, so for this range we will only count 9.
Total unique numbers with 2s in 100-199 = 19.

For 5s: Just like above we will get 19.
But in these 19, we have 125 and 152 - which are already counted in 2s above, and will be double counted in 5s again.

So unique numbers in 100-199 having 2 or 5 = 19 + 19 - 2 = 36.

We have 100-199, 300-399, 400-499, 600-699, 700-799, 800-899, 900-999 = 7 such sets of 100 numbers.
So total numbers having 2 or 5 = (2 x 100) + (36 x 7) = 200 + 252 = 452

Numbers not having 2 or 5 = 900 - 452 = 448

How many of the integers from 100 to 999, inclusive, do NOT contain ei

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many of the integers from 100 to 999, inclusive, do NOT contain ei

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards