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# Find the number of natural numbers which lie between 10^7 and 10^8

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SVP
Joined: 20 Jul 2017
Posts: 1506
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Find the number of natural numbers which lie between 10^7 and 10^8  [#permalink]

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24 Mar 2020, 07:54
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3
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Difficulty:

55% (hard)

Question Stats:

50% (02:22) correct 50% (02:04) wrong based on 20 sessions

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Find the number of natural numbers which lie between 10^7 and 10^8 and which have products of their digits as 6?

A. 36
B. 55
C. 64
D. 81
E. 105
DS Forum Moderator
Joined: 19 Oct 2018
Posts: 1851
Location: India
Re: Find the number of natural numbers which lie between 10^7 and 10^8  [#permalink]

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24 Mar 2020, 09:20
1
6= 1*1*1*1*1*1*1*6 = 1*1*1*1*1*1*2*3

Case 1- When 1 of the 8 digits is '6' and remaining are '1'

Total possible arrangements =$$\frac{ 8!}{1!*7! }= 8$$

Case 2 - When two of the 8 digits are '2' and '3' and the remaining are '1'.

Total possible arrangements =$$\frac{ 8!}{1!*1!*6!}$$ = 56

Total positive integers which lie between 10^7 and 10^8 and which have products of their digits as 6 = 8+56 = 64

Dillesh4096 wrote:
Find the number of natural numbers which lie between 10^7 and 10^8 and which have products of their digits as 6?

A. 36
B. 55
C. 64
D. 81
E. 105
Intern
Joined: 01 Apr 2020
Posts: 6
Re: Find the number of natural numbers which lie between 10^7 and 10^8  [#permalink]

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02 Apr 2020, 05:07
Use Mississippi rule for repeated digits

Since only 4 Numbers can be used in this 8 digit number, 1,2,3,6 and that also in only 2 ways possible since product of all digits has to be 6

11111116, 11111123

In cases where order matters
Number of ways to rearrange 1st number = 8!7! (Mississippi rule)
Number of ways to rearrange 2nd number = 8!/6! ( Same rule)

Re: Find the number of natural numbers which lie between 10^7 and 10^8   [#permalink] 02 Apr 2020, 05:07