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# How many times will the digit 7 be written?

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Joined: 28 Mar 2017
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Re: How many times will the digit 7 be written?  [#permalink]

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10 Mar 2018, 05:24
seekmba wrote:
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

_ _ _

1. When all are 7, we have 1 combination i.e. 7 7 7 and since 7 appears 3 times we have 1*3= 3

2. When we have 2 7s, 7 7 _, we can fill the dash with 9 remaining digits and these can be further arranged in 3 ways.
Total ways = 9*3=27; but 7 appears 2 times in each combination; thus we have 27*2 = 54

3. When we have 1 7; 7 _ _ , we can fill the remaining dashes with remainng 9 digits.
Total ways = 9*9*3= 243

Thus Final Total = 243+54+3 = 300
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Re: How many times will the digit 7 be written?  [#permalink]

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12 Mar 2018, 07:26
Top Contributor
seekmba wrote:
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

Here's one way to look at it.
Write all of the numbers as 3-digit numbers.
That is, 000, 001, 002, 003, .... 998, 999

NOTE: Yes, I have started at 000 and ended at 999, even though though the question asks us to look at the numbers from 1 to 1000. HOWEVER, notice that 000 and 1000 do not have any 7's so the outcome will be the same.

First, there are 1000 integers from 000 to 999
There are 3 digits in each integer.
So, there is a TOTAL of 3000 individual digit. (since 1000 x 3 = 3000)

Each of the 10 digits is equally represented, so the 7 will account for 1/10 of all digits.

1/10 of 3000 = 300

So, there are 300 0's, 300 1's, 300 2's, 300 3's, . . ., and 300 9's in the integers from 000 to 999

Cheers,
Bren
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How many times will the digit 7 be written?  [#permalink]

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11 Sep 2018, 16:21
seekmba wrote:
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

The digit 7 as the hundreds digit appears 100 times (700 to 799). As the tens digit, it appears 100 times also (ten times each in the 70s, 170s, 270s, …, 970s). As the units digit, it appears 100 times also (7, 17, 27, …, 997). Therefore, the digit 7 has appeared a total of 300 times.

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How many times will the digit 7 be written?  [#permalink]

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30 Nov 2018, 02:58
Solution using permutation combination:

Digits needed to form numbers: 0,1,2,3,4,5,6,7,8,9 (10 digits)

Case 1: 7 comes in exactly one position:

7 _ _ , _ 7 _ and _ _ 7
For the above three sub cases, the number of possibilities are (1 x 9 x 9, 9 x 1 x 9 and 9 x 9 x 1) which is 81 for each sub case.
Total number of 7s : 3 x 81 = 243
Note: (a) 9 possibilities (0-9, excluding 7)
(b) This covers 2-digit 7-numbers as well since we have included 0.

Case 2: 7 comes in exactly 2 positions:

77_ , 7_7 and _77
For the above sub cases the number of possibilities are: (1x1x9 , 1x9x1 , 9x1x1)
Each sub case has 9 possibilities. But the question asks about the number of 7s so each sub case will have 2x9 = 18 possibilities.
Total number of 7s = 3x18 = 54

Case 3: 7 comes in all places:

777 : 1 possibility ; Total number of 7s=3

Adding 243 +54 +3 = 300

If you like the explanation, then don't forget to give kudos.
How many times will the digit 7 be written? &nbs [#permalink] 30 Nov 2018, 02:58

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