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How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 49% (02:44) correct 51% (02:34) wrong based on 110 sessions

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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

*kudos for all correct solutions

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Math Expert V
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Posts: 8331
How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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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

*kudos for all correct solutions

Single digit...3 -1
2-digits ...30-39 - 10
3-digits.....300-399 -100
Total 1+19+100=111

Let's see how many end with 3....
Every tens has one 3.. 3,13,23,33...,93-10
Number of tens ... 500/10=50

Numbers repeated above .. those starting and ending with 3
10 in 300s - 303,313,...393
33 in 2-digits
3 in single digit
So total 12

Ans 111+50-12=161-12=149

A
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Re: How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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chetan2u Thanks a lot for the easy explanation. I just have a quick question, the no of digits between 30 and 39 that start with 3 would be 10 right? which makes the total 111? Also, there would be one more redundant value in the single digit group which is 3? Is that correct or am I missing something?
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Re: How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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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

*kudos for all correct solutions

Lets start with single digit numbers, we have only 1 number 3

Now 2 digit numbers we have of the form 3_ & _3
3_ are 10 numbers (30,31,32,33....39)
_3 are 9 numbers (13,23,33,.....93)
we get 33 in both sets, hence we remove 1 from total of 2 digit numbers,

Hence total 2 digit numbers is 18

Now 3 digit numbers, we have of the form 3 _ _ & _ _ 3
3 _ _ are 100 numbers (300,301,302,303,.....399)
_ _ 3 are 40 numbers (103,113,....203,213,....303,313,....413,423,...493)
we also have numbers of the form 3 _ 3 (303,313,...393) common to both sets, these are 10 such numbers.

Hence total 3 digit numbers is 100 + 40 -10 = 130

Hence the total numbers of the required form = 1 + 18 + 130 = 149.

Thanks,
GyM

Ps. You can also use combinatorics to count the numbers, for e.g. 3 _ _ can be 1*10C1*10C1 = 100 & _ _ 3 can be 4C1*10C1*1 = 40
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GMAT 1: 560 Q42 V25 GMAT 2: 570 Q43 V27 GMAT 3: 710 Q49 V39 How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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The total number of 3 digit numbers that begin with 3 = 100.
The total number of numbers that end with 3 = From 0 to 100 - 10
From 100 to 200 - 10
From 200 to 300 - 10, Excluding all numbers ending with 3 in 300s as already counted.
From 400 to 500 - 10
The total number of numbers beginning with 3 but are 2 digit numbers = 9 (30, 31, 32, 34, 35, 36, 37, 38, 39) - Excluding 33 as was already counted in the ending with 3 section.
Total = 100 + 40 + 9 = 149.
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How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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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

*kudos for all correct solutions

We need to include one digit, two-digit, and three-digit numbers less than 500.

1-99
3 is the only one digit number
30-39 are the two digit numbers starting with 3. (Total of 10 numbers)
13,23,43,53,63,73,83,93 are the two-digit numbers ending with 3. (Total of 8 numbers)

$$100-199$$
113,123,133,143,153,163,173,183,and 193 are the two-digit numbers ending with 3 in this range

Similarly, both the ranges 200-299 and 400-499 have 10 numbers each which end with 3.

The range $$300-399$$ has all the hundred numbers starting with 3(we needn't include those
numbers in this range which end with 3 because that would lead to double counting)

Therefore, there are a total of $$1 + 8 + 10*4 + 100 = 149$$(Option A) such numbers less than 500.
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How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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rishabhmail125 wrote:
chetan2u Thanks a lot for the easy explanation. I just have a quick question, the no of digits between 30 and 39 that start with 3 would be 10 right? which makes the total 111? Also, there would be one more redundant value in the single digit group which is 3? Is that correct or am I missing something?

Yes, rishabhmail125 - Your observation is correct.

The numbers starting with 3 will include 33 in the list. The total numbers starting with 3 is 111.
Also, 3 is part of both the lists. The total numbers which are part of both lists are 12 in number.

Therefore, the total numbers(starting with 3, ending with 3, or both) less than 500 are 149(111+50-12)
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Re: How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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rishabhmail125 wrote:
chetan2u Thanks a lot for the easy explanation. I just have a quick question, the no of digits between 30 and 39 that start with 3 would be 10 right? which makes the total 111? Also, there would be one more redundant value in the single digit group which is 3? Is that correct or am I missing something?

Rishabh, you are absolutely correct..
There was a typo in the post, edited now.
There will be 111 positive integers starting with 3 and 50 ending with 3 but 12 common to both so 111+50-12=149
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Re: How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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Top Contributor
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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

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

Cheers,
Brent
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Re: How many positive integers less than 500 begin with a 3, end with a 3,  [#permalink]

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Sharing my approach to this question using the slot method:

In all there are $$10$$ digits from $$0, 1, 2, 3, ... , 9$$ inclusive

Single Digit Number $$= 1$$
Only $$3$$

Double Digit Numbers $$= 18$$
a. Starts and ends with 3 $$=$$ Only $$33$$
b. Starts with 3 but ends with a number other than 3 $$=$$ 1 9 $$= 9$$
c. Starts with a number other than 3 but ends with 3 $$=$$ 8 1 $$= 8$$ (The first digit cannot be 0 and 3, hence 8)

Three Digit Numbers $$= 130$$
a. Starts and ends with 3 $$=$$ 1 10 1
b. Starts with 3 but ends with a number other than 3 $$=$$ 1 10 9 $$= 90$$
c. Starts with a number other than 3 but ends with 3 $$=$$ 3 10 1 $$= 30$$ (Only 1, 2 and 4 can take the first place as the number has to be between 100 and 500)

Combining the results = 1 + 18 + 130 = 149 (Ans. A)
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Cheers. Wishing Luck to Every GMAT Aspirant! Re: How many positive integers less than 500 begin with a 3, end with a 3,   [#permalink] 20 Jan 2020, 08:47
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