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# How many license plates are possible if the plate must contain exactly

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Math Expert
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How many license plates are possible if the plate must contain exactly [#permalink]

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13 May 2017, 14:26
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How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049
[Reveal] Spoiler: OA

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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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13 May 2017, 20:31
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Bunuel wrote:
How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049

The license plate forms as $$abcde$$ with $$a \neq 0$$

With $$a$$, we can select 9 different digits from 1 to 9.
With $$b$$, we can select 9 remaining digits from 0 to 9 (1 digit is picked out with $$a$$)
With $$c$$, we can select 8 remaining digits from 0 to 9 (2 digits are picked out)
With $$d$$, we can select 7 remaining digits from 0 to 9 (3 digits are picked out)
With $$e$$, we can select 6 remaining digits from 0 to 9 (4 digits are picked out)

The total combination is $$9 \times 9 \times 8 \times 7 \times 6 =27,216$$

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Math Expert
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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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13 May 2017, 20:38
Bunuel wrote:
How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049

Hi...

Go stepwise and don't go wrong on the first step..
1) the first digit can be anything except 0, so 9 ways.
2) second can be remaining 8 from step 1 and 0, so 8+1=9 ways.
3) third digit remaining 8 from step 2... 8ways
4) fourth...7 ways
5) fifth....6 ways..

Total 9*9*8*7*6
Number should be EVEN but not multiple of 10.. ONLY A and C.
Number should be multiple of 9 .... Add the digits to find
A. 8238..8+2+3+8=21.... Not a multiple of 9... Out
C. 27,216....2+7+2+1+6=18..... Multiple of 9.. ANS

C
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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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13 May 2017, 22:40
Bunuel wrote:
How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049

METHOD-1

Plate Number = _ _ _ _ _

Choices for the leftmost places = 9 (any digit from 1 to 9)
i.e. Plate Number = 9 _ _ _ _

Choices for the second from leftmost places = 9 (any digit from 0 to 9 except the one used for leftmost place)
i.e. Plate Number = 9 * 9 _ _ _

Choices for the Third from leftmost places = 8 (any digit from 0 to 9 except the two digits used for left two place)
i.e. Plate Number = 9 * 9 * 8 _ _

Choices for the Forth from leftmost places = 7 (any digit from 0 to 9 except the two digits used for left three place)
i.e. Plate Number = 9 * 9 * 8 * 7 _

Choices for the Forth from leftmost places = 6 (any digit from 0 to 9 except the two digits used for left Four place)
i.e. Plate Number = 9 * 9 * 8 * 7 * 6

So Total number plates = 27216

METHOD-2

Plate Number = _ _ _ _ _

Choices for the leftmost places = 9 (any digit from 1 to 9)
i.e. Plate Number = 9 _ _ _ _

Other 4 digits may be any 4 digits out of remaining 9 digits which can be chosen in 9C4 ways and can be arranged in 4! ways

So total Number plates = 9* 9C4*4! = 27216

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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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14 May 2017, 14:54
We have only ten choices of digits and five placeholders to fill in, however, the first place can not be 0.

First place can have 9 choices, second place can have again 9 Choices (since we couldn't put zero in first place, one digit is already gone and we are left with zero and 8 more digits bringing our digits back to 9), third spots 8 choices, fourth spots 7 choices and fifth spot 6 choices, as shown below:

Therefore, $$9*9*8*7*6=27216$$

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How many license plates are possible if the plate must contain exactly [#permalink]

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16 May 2017, 00:04
5 digits arranged from 0 to 9 when digits can not be repeated and 0 can not take the first place is done in 9*9*8*7*6 ways .. If you notice the answer options closely , you may notice all options have different unit digits . so find the unit digit of the above arrangement which is 6. Answer (C) is satisfies this.

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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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18 May 2017, 20:08
1
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Expert's post
Bunuel wrote:
How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049

Since we can’t use 9 as the first digit, we have 9 choices for the first digit. Since we can’t repeat the digits, we have 9 choices for the second digit, 8 for the third, 7 for the fourth, and 6 for the fifth.

Thus, there are 9 x 9 x 8 x 7 x 6 = 27,216 ways to create the license plates.

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Re: How many license plates are possible if the plate must contain exactly [#permalink]

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02 Sep 2017, 13:23
Bunuel wrote:
How many license plates are possible if the plate must contain exactly five digits, and the plate cannot start with 0 or repeat any digits? (The digits must be in a row and the plate contains no other characters.)

A. 8,238
B. 15,120
C. 27,216
D. 30,240
E. 59,049

There's 9 possibilities for the first slot (1-9) then there's 9 possibilities for the second slot (0-8) and then 8 possibilities for the third slot, 7 for the fourth and 6 for the 5th slot

C

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Re: How many license plates are possible if the plate must contain exactly   [#permalink] 02 Sep 2017, 13:23
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