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Since 2001, the standard serial numbers on a New York state license pl 30 Dec 2014, 07:38
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Tough and Tricky questions: Combinations.



Since 2001, the standard serial numbers on a New York state license plate are 3 letters followed by 4 digits. How many different license plates are possible if letters and digits can be repeated?

A. 26 × 3 × 10 × 4
B. 26 × 25 × 24 × 10 × 9 × 8 × 7
C. 26³ × 9 × 9 × 9 × 9
D. 26 × 25 × 24 × 10 000
E. 26³ × 10 000

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Since 2001, the standard serial numbers on a New York state license pl 30 Dec 2014, 07:57
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Answer E.

Since the 7 positions are completely independent from each other, we simply need to multiply the possibilities for each of the 7 positions. Each of the three letters allows 26 combinations (assuming 26 different letters) and the digits allow 10 different combinations for each position. We therefore get:

\(26 * 26 * 26 * 10 * 10 * 10 * 10 = 26^3 * 10 ^4 = 26 ^3 * 10,000\)
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Since 2001, the standard serial numbers on a New York state license pl 30 Dec 2014, 22:25
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Solution:
3 repeated letters can be chosen as 26 * 26 * 26
4 repeated digits can be chosen as 10 * 10 * 10 * 10
Answer E
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Since 2001, the standard serial numbers on a New York state license pl 08 Jan 2015, 08:18
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Bunuel

Tough and Tricky questions: Combinations.



Since 2001, the standard serial numbers on a New York state license plate are 3 letters followed by 4 digits. How many different license plates are possible if letters and digits can be repeated?

A. 26 × 3 × 10 × 4
B. 26 × 25 × 24 × 10 × 9 × 8 × 7
C. 26³ × 9 × 9 × 9 × 9
D. 26 × 25 × 24 × 10 000
E. 26³ × 10 000

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OFFICIAL SOLUTION:

(E) The formula for permutations of events is the product of the number of ways each event can occur. There are 26 letters and 10 digits. So there are 26 × 26 × 26 options for the three letters, and 10 × 10 × 10 × 10 for the four digits. The number of different license plates is 26 × 26 × 26 × 10 × 10 × 10 × 10 = 26³ × 10 000.

The correct answer is choice (E).
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Since 2001, the standard serial numbers on a New York state license pl 30 Jun 2018, 04:52
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Would the answer be A if the letters and numbers were not allowed to be repeated
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Since 2001, the standard serial numbers on a New York state license pl 30 Jun 2018, 10:32
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Would the answer be A if the letters and numbers were not allowed to be repeated

In this case B would be correct: (26 × 25 × 24) × (10 × 9 × 8 × 7)
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Since 2001, the standard serial numbers on a New York state license pl 09 Aug 2019, 22:42
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Bunuel
Bunuel

Tough and Tricky questions: Combinations.



Since 2001, the standard serial numbers on a New York state license plate are 3 letters followed by 4 digits. How many different license plates are possible if letters and digits can be repeated?

A. 26 × 3 × 10 × 4
B. 26 × 25 × 24 × 10 × 9 × 8 × 7
C. 26³ × 9 × 9 × 9 × 9
D. 26 × 25 × 24 × 10 000
E. 26³ × 10 000

Kudos for a correct solution.

OFFICIAL SOLUTION:

(E) The formula for permutations of events is the product of the number of ways each event can occur. There are 26 letters and 10 digits. So there are 26 × 26 × 26 options for the three letters, and 10 × 10 × 10 × 10 for the four digits. The number of different license plates is 26 × 26 × 26 × 10 × 10 × 10 × 10 = 26³ × 10 000.

The correct answer is choice (E).

Hi,

I have a slight confusion about this problem.

Since all the answers are permutations, when we say 26 x 26 x 26 x 10 x 10 x 10 x 10 aren't we saying that order matters ?

What I mean to say is that there might very well be a number plate with CLL 1234 and as two Ls will be taken as different under this permutation based method the total count will include another CLL 1234, whereas hey should be counted as just 1 (2 number plates can not be the same).

Shouldn't we make an adjustment for this ?

Will appreciate a response on this please.
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Since 2001, the standard serial numbers on a New York state license pl 04 Sep 2019, 13:20
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Quote:

Hi,

I have a slight confusion about this problem.

Since all the answers are permutations, when we say 26 x 26 x 26 x 10 x 10 x 10 x 10 aren't we saying that order matters ?

What I mean to say is that there might very well be a number plate with CLL 1234 and as two Ls will be taken as different under this permutation based method the total count will include another CLL 1234, whereas hey should be counted as just 1 (2 number plates can not be the same).

Shouldn't we make an adjustment for this ?

Will appreciate a response on this please.

altairahmad
Hello, The question is asking how many combinations are possible in total. We are not concerned whether the number plates are same or not. So Lets say AAA0000 next will be AAA0001 next will be AAA0002 and so on.. So All we need to figure out is what is the number of exhaustive cases.
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Since 2001, the standard serial numbers on a New York state license pl 18 May 2022, 03:29
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