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All the possible five-digit numbers are formed using only the digits

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Manager
Joined: 18 Jul 2019
Posts: 54
All the possible five-digit numbers are formed using only the digits  [#permalink]

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25 Nov 2019, 01:20
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Difficulty:

55% (hard)

Question Stats:

46% (02:11) correct 54% (02:12) wrong based on 13 sessions

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All the possible five-digit numbers are formed using only the digits 4, 5 and 6. How many of them will have at least two fives?

(A) 131
(C) 195
(B) 219
(D) 230
(E) 250
Director
Joined: 27 May 2012
Posts: 945
All the possible five-digit numbers are formed using only the digits  [#permalink]

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25 Nov 2019, 08:17
CaptainLevi wrote:
All the possible five-digit numbers are formed using only the digits 4, 5 and 6. How many of them will have at least two fives?

(A) 131
(C) 195
(B) 219
(D) 230
(E) 250

Total cases $$(3)^4$$ - ( Ways to get no five + ways to get one five ) = Ways to get at least two five.

No five's =2*2*2*2*2 = 32

Number of ways to get ONLY one 5:
One five and rest all 4 ( 5, 4, 4, 4, 4) = $$\frac{5!}{4!}= 5$$ ways
One five and rest all 6 ( 5, 6, 6, 6, 6) = $$\frac{5!}{4!}= 5$$ ways
One five ,three 6 and one 4 ( 5, 6, 6, 6, 4) =$$\frac{5!}{3!}=20$$ ways
One five ,three 4 and one 6( 5, 4, 4, 4, 6)=$$\frac{5!}{3!}$$=20 ways
One five ,two 6 and two 4 ( 5 6 6 4 4) =$$\frac{5!}{2!*2!}$$=30 ways

So total ways=$$(3)^4$$=243
No fives in = 32 ways
One five in = 5+5+20+20+30 =80 ways
So 243 - (32+80)= 131 Ways to get at least two five's.

Ans- A
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- Stne
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Joined: 19 Jan 2019
Posts: 11
Re: All the possible five-digit numbers are formed using only the digits  [#permalink]

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26 Nov 2019, 03:02
any other way to solve this ?
Intern
Joined: 19 Jan 2019
Posts: 11
Re: All the possible five-digit numbers are formed using only the digits  [#permalink]

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26 Nov 2019, 03:05
how is it 3 raise to the power 4 ?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9859
Location: Pune, India
Re: All the possible five-digit numbers are formed using only the digits  [#permalink]

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26 Nov 2019, 05:11
1
CaptainLevi wrote:
All the possible five-digit numbers are formed using only the digits 4, 5 and 6. How many of them will have at least two fives?

(A) 131
(C) 195
(B) 219
(D) 230
(E) 250

Five digit number: ____ ____ ____ ____ ____

Each blank can be filled up in 3 ways. So there are 3 * 3 * 3 * 3 * 3 = 243 ways of making the five digit number.

In how many of them will there be no 5? There will be no 5 in numbers made from only 4 and 6. So each spot will have only 2 options in that case.
Then there are 2 * 2 * 2 * 2 * 2 = 32 ways of making the five digit number.

In how many of them will there be only one 5? Select 1 spot for the 5 out of 5 spots in 5 ways.
The number of ways of filling up each of the other 4 spots is 2.
Total numbers = 5 * 2 * 2 * 2 * 2 = 80 ways of making the five digit number.

So in 32 + 80 = 112 numbers, there will be no 5 or only one 5.

In the other 243 - 112 = 131 numbers, there will be at least two 5s.

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Karishma
Veritas Prep GMAT Instructor

Re: All the possible five-digit numbers are formed using only the digits   [#permalink] 26 Nov 2019, 05:11
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