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# How many times will the digit 7 be written when listing from

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CEO
Joined: 21 Jan 2007
Posts: 2760
Location: New York City
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How many times will the digit 7 be written when listing from [#permalink]  15 Oct 2007, 11:24
How many times will the digit 7 be written when listing from 1 to 1000?

110
111
271
300
304
Director
Joined: 22 Aug 2007
Posts: 569
Followers: 1

Kudos [?]: 16 [0], given: 0

Re: Counting [#permalink]  15 Oct 2007, 11:50
bmwhype2 wrote:
How many times will the digit 7 be written when listing from 1 to 1000?

110
111
271
300
304

D,

we have 3 options for numbers between 1 and 1000

one digit: 0, 1,2,3......9.
two digits: 10,11........99
three digits: 100.........999

1000 does not have 7, so no need to count it.

one digit: only 1 number
two digits: 1*10+9*1=19 numbers

there are 1*10 two digit numbers that have 7 as first digit and any other (from 0 to 9 digit) as second digit. And there are 9*1 two digit numbers that have any other (from 1 to 9 digit) as first digit and 7 as second digit.
Zero can not be first intenger.

three digits: 1*10*10+9*1*10+9*10*1=280 numbers

same logic as for two digit numbers.

The total number of 7`s that will be written is

1+19+280=300.

I got this question wrong cuz i mis read the question....i counted numbers from 1-100 that have digit 7, (in this case you would have to minus repeatitions ) but the questions says "many times will the digit 7 be written", thus no need to minus repeatition.
Senior Manager
Joined: 04 Jun 2007
Posts: 374
Followers: 1

Kudos [?]: 22 [0], given: 0

[#permalink]  15 Oct 2007, 12:08
numbers where 7 is unit digit between 1-1000= 10*10= 100
numbers where 7 is tenth digit= 10*10= 100
numbers where 7 is hundredth digit= 799-700+1= 100

total= 100+100+100= 300

dunno if there is an easier/faster way
Current Student
Joined: 08 Oct 2007
Posts: 169
Location: Berkeley, CA
Schools: Berkeley-Haas MBA
WE 1: Investment Management (fund of funds)
WE 2: Private Equity ($2bn generalist fund) Followers: 1 Kudos [?]: 17 [0], given: 7 Re: Counting [#permalink] 16 Oct 2007, 06:13 bmwhype2 wrote: How many times will the digit 7 be written when listing from 1 to 1000? 110 111 271 300 304 all singles digits = 10 times 9 all doubles digits = 10 times 9 all hundreds digits = 100 add them up and I get 280. I don't get this question. do we count 777 as 3 counts of 7?? Current Student Joined: 08 Oct 2007 Posts: 169 Location: Berkeley, CA Schools: Berkeley-Haas MBA WE 1: Investment Management (fund of funds) WE 2: Private Equity ($2bn generalist fund)
Followers: 1

Kudos [?]: 17 [0], given: 7

Re: Counting [#permalink]  16 Oct 2007, 06:17
stopper5 wrote:
bmwhype2 wrote:
How many times will the digit 7 be written when listing from 1 to 1000?

110
111
271
300
304

all singles digits = 10 times 9

all doubles digits = 10 times 9

all hundreds digits = 100

add them up and I get 280.

I don't get this question. do we count 777 as 3 counts of 7??

ok, my bad, I should have multiplied by 10
Senior Manager
Joined: 04 Jun 2007
Posts: 374
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Kudos [?]: 22 [0], given: 0

[#permalink]  16 Oct 2007, 06:41
btw, this will hold true for any digit between 1-9.

if the question was-

How many times will the digit 3 be written when listing from 1 to 1000?

again it will be 300.
VP
Joined: 08 Jun 2005
Posts: 1146
Followers: 6

Kudos [?]: 136 [0], given: 0

[#permalink]  16 Oct 2007, 11:24
How many times will the digit 7 be written when listing the integers from 1 to 1000?

A) 110
B) 111
C) 271
D) 300
E) 304

N = 1,000

When 7 appears three times (i.e. 777)

P11 = 1/10*1/10*1/10 = 1/1000

P1 = 1/1000*3 = 3/1000 (since we are counting the appearances of sevens!)

When 7 appears twice (i.e. X77 or 77X or 7X7)

P21 = 1/10*1/10*9/10 = 9/1000
P22 = 1/10*1/10*9/10 = 9/1000
P23 = 1/10*1/10*9/10 = 9/1000

1/10 is for picking 7 out of 10 digits and the 9/10 is for picking any other digit except 7.

P2 = 27/1000*2 = 54/1000 (since we are counting the appearances of sevens!)

P31 = 1/10*9/10*9/10 = 81/1000
P32 = 9/10*1/10*1/10 = 81/1000
P33 = 9/10*9/10*1/10 = 81/1000

P3 = 243/1000*1 = 243/1000 (since we are counting the appearances of sevens!)

When 7 appears once (i.e. XX7 or 7XX or X7X)

Sum of P = 3/1000+54/1000+243/1000 = 300/1000

N*P = 1000*300/1000 = 300

The answer is (D)
Manager
Joined: 07 Sep 2007
Posts: 121
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Kudos [?]: 13 [0], given: 0

Re: Counting [#permalink]  16 Oct 2007, 15:03
bmwhype2 wrote:
How many times will the digit 7 be written when listing from 1 to 1000?

110
111
271
300
304

XXX7 => XXX = 0-99 = 100 times
XX7X => XXX = 0-99 = 100 times
X7XX => XXX = 0-99 = 100 times

Sum it all up, 300.
Director
Joined: 03 May 2007
Posts: 888
Schools: University of Chicago, Wharton School
Followers: 6

Kudos [?]: 74 [0], given: 7

Re: Counting [#permalink]  16 Oct 2007, 22:12
JingChan wrote:
bmwhype2 wrote:
How many times will the digit 7 be written when listing from 1 to 1000?

110
111
271
300
304

XXX7 => XXX = 0-99 = 100 times
XX7X => XXX = 0-99 = 100 times
X7XX => XXX = 0-99 = 100 times

Sum it all up, 300.

excellent.

unit digit (xx7): 0-99 = 10 x 10 = 100
tens digit (x7x): 70-79 = 10 x 10 = 100
hundred digit (7xx) : 700-799 = 100
sum = 300
Re: Counting   [#permalink] 16 Oct 2007, 22:12
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