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# A small nation issues license plates that consist of just one number

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Math Expert
Joined: 02 Sep 2009
Posts: 56269
A small nation issues license plates that consist of just one number  [#permalink]

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03 Jan 2019, 09:10
1
7
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Difficulty:

95% (hard)

Question Stats:

39% (02:25) correct 61% (02:17) wrong based on 118 sessions

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A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

(A) 1,599,990
(B) 1,599,999
(C) 1,600,000
(D) 4,569,759
(E) 4,569,760

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Joined: 12 Dec 2017
Posts: 79
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Schools: Yale '19, GMBA '20, XLRI
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Re: A small nation issues license plates that consist of just one number  [#permalink]

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03 Jan 2019, 22:05
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2
A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

Total slots = 5 [ digit, letter, letter, letter, letter ]

slot 1 : 10 possibilities (0-9)
slot 2 : 20 possibilities (20 letters)
slot 3: 20 possibilities (20 letters : repetition allowed)
slot 4: 20 possibilities
slot 5 : 20 possibilities

total ways= 10* (20)^4

disallowed combinations: 10*1

10* [ (20)^4 -1 ]
10*159999 = 1599990

A
Manager
Joined: 12 Dec 2017
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Re: A small nation issues license plates that consist of just one number  [#permalink]

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03 Jan 2019, 22:08
1
kanakdaga wrote:
A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

Total slots = 5 [ digit, letter, letter, letter, letter ]

slot 1 : 10 possibilities (0-9)
slot 2 : 20 possibilities (20 letters)
slot 3: 20 possibilities (20 letters : repetition allowed)
slot 4: 20 possibilities
slot 5 : 20 possibilities

total ways= 10* (20)^4

disallowed combinations: 10*1

10* [ (20)^4 -1 ]
10*159999 = 1599990

A

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Re: A small nation issues license plates that consist of just one number  [#permalink]

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06 Jan 2019, 05:46
Bunuel wrote:
A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

(A) 1,599,990
(B) 1,599,999
(C) 1,600,000
(D) 4,569,759
(E) 4,569,760

1 no would be 10
and 4 letters 20^4
so total 10*20^4 = 16*10^5
one combination of 4 letter not allowed to appear on license plates = 1^4 * 10 = 10

so left with valid no= 16*10^5-10 = 159990 IMO A
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Joined: 22 Jan 2014
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Schools: Johnson '21 (WL)
Re: A small nation issues license plates that consist of just one number  [#permalink]

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11 Jan 2019, 05:27
Bunuel wrote:
A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

(A) 1,599,990
(B) 1,599,999
(C) 1,600,000
(D) 4,569,759
(E) 4,569,760

I was confused with the wording one four letter combination is not allowed
Lets say it was
FUKC
Did they mean
FKUC and other combinations not allowed too.
Intern
Joined: 01 May 2016
Posts: 1
Re: A small nation issues license plates that consist of just one number  [#permalink]

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20 Jan 2019, 15:17
I think the question stem needs to be a bit clearer. It does not specify that the numerical digit can only appear in one position!
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Joined: 05 Jan 2018
Posts: 8
Re: A small nation issues license plates that consist of just one number  [#permalink]

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29 Jan 2019, 14:01
KomalSg wrote:
Bunuel wrote:
A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

(A) 1,599,990
(B) 1,599,999
(C) 1,600,000
(D) 4,569,759
(E) 4,569,760

I was confused with the wording one four letter combination is not allowed
Lets say it was
FUKC
Did they mean
FKUC and other combinations not allowed too.

The combination remains same, in your case "FKUC" however this is to be suffixed with a digit and that digit can be any between 0 to 9. Hence. 10 options to be subtracted.
Re: A small nation issues license plates that consist of just one number   [#permalink] 29 Jan 2019, 14:01
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