Challenge: How many positive integers less than 500 begin

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How many positive integers less than 500 begin with a 3, end with a 3, or both?
A) 149
B) 150
C) 159
D) 199
E) 200

Answer: A
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by arosman » Wed Jun 20, 2018 10:04 am
Just to make things organized I broke down 1-500 into 5 sets:

[1-99]
------> 3, 13, 23, 43, 53, 63, 73, 83, 93 + 30-39 ------------> 9 + 10 = 19 numbers


[100 - 199)
-----> 103, 113, 123.....193 --------> 10 numbers


[ 200 - 299]
----> 203, 213, 223.....293 -------> 10 numbers


[300 - 399]
--------> All numbers begin with 3 ----> 100 numbers


[400 - 499]
---> 403, 413, 423....493 --------> 10 numbers

Total numbers that begin or end in 3 ---> 100 + 10 + 10 + 10 + 19 = 149

Answer: A
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by Brent@GMATPrepNow » Fri Jun 22, 2018 5:11 am
Brent@GMATPrepNow wrote:How many positive integers less than 500 begin with a 3, end with a 3, or both?
A) 149
B) 150
C) 159
D) 199
E) 200
Here's a different approach:

There are 499 integers from 1 to 499 inclusive.

Let's determine how many of those 499 integers DO NOT meet the condition of beginning with a 3, ending with a 3, or both

1-digit integers that DO NOT begin with a 3, end with a 3, or both
1, 2, 4, 5, 6, 7, 8, and 9
TOTAL = 8

2-digit integers that DO NOT begin with a 3, end with a 3, or both
In the tens position, we can have 1, 2, 4, 5, 6, 7, 8, or 9 (8 possibilities)
In the units position, we can have 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 possibilities)
TOTAL number of 2-digit integers that DO NOT begin with a 3, end with a 3, or both = (8)(9) = 72

3-digit integers that DO NOT begin with a 3, end with a 3, or both
In the hundreds position, we can have 1, 2, or 4 (3 possibilities)
In the tens position, we can have 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 (10 possibilities)
In the units position, we can have 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 possibilities)
TOTAL number of 3-digit integers that DO NOT begin with a 3, end with a 3, or both = (3)(10)(9) = 270

So, the TOTAL number of integers that DO NOT meet the given condition = 8 + 72 + 270 = 350

TOTAL number of integers that MEET the given condition
So, the = 499 - 350 = 149

Answer: A

Cheers,
Brent
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