Math Destroyer 2017 #39

[palmtreesarecool]

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A license plate must be made of any three digits followed by a letter. How many different license plates is possible if each digit can only be used once and the first digit cannot be 0?
A. 9x9x8x26
B. 10x9x8x26
C. 9x8x7x26
D. 9x10x8x26
E. 9x9x7x26

the solutions say the answer is A, but why isn't the answer C?
Thank you in advance 🙂

 

[DAT Destroyer]

A license plate must be made of any three digits followed by a letter. How many different license plates is possible if each digit can only be used once and the first digit cannot be 0?
A. 9x9x8x26
B. 10x9x8x26
C. 9x8x7x26
D. 9x10x8x26
E. 9x9x7x26

the solutions say the answer is A, but why isn't the answer C?
Thank you in advance 🙂

Since it's 3 digits, 0 can't be the first digit which leaves us with 9 choices for the first digit. Because no digits can repeat, we have 9 possibilities left for the second spot. And for the third spot 8 digits left and for the last spot 26 possibilities left.
Hence the answer is:
9*9*8*26

 

[palmtreesarecool]

Since it's 3 digits, 0 can't be the first digit which leaves us with 9 choices for the first digit. Because no digits can repeat, we have 9 possibilities left for the second spot. And for the third spot 8 digits left and for the last spot 26 possibilities left.
Hence the answer is:
9*9*8*26

Why do you have 9 possibilities left for the second spot? If you have 9 choices for the first digit and no digits can repeat, wouldn't it make sense that you would have 8 possibilities left for the second spot? You would use one of the 9 choices for the first spot and then you would have 8 possibilities left over for the second spot since none of the digits can repeat. I'm sorry; I just don't understand the reasoning behind having 9 possibilities for the second spot.

 

[KkOma]

Why do you have 9 possibilities left for the second spot? If you have 9 choices for the first digit and no digits can repeat, wouldn't it make sense that you would have 8 possibilities left for the second spot? You would use one of the 9 choices for the first spot and then you would have 8 possibilities left over for the second spot since none of the digits can repeat. I'm sorry; I just don't understand the reasoning behind having 9 possibilities for the second spot.

So i think its because since the first digit cant be 0, you have 9 possibilities (1-9), now you can include 0 for the second spot and you already used up one of the 1-9 possibilities, so you would have 8, but include the 0, now its 9. now you used up two digits, and they can't repeat, so you have 8 for the third, and 26 to finish it off.

 

[palmtreesarecool]

So i think its because since the first digit cant be 0, you have 9 possibilities (1-9), now you can include 0 for the second spot and you already used up one of the 1-9 possibilities, so you would have 8, but include the 0, now its 9. now you used up two digits, and they can't repeat, so you have 8 for the third, and 26 to finish it off.

That makes so much sense; thank you very much! I appreciate your explanation.

 

[DAT Destroyer]

That makes so much sense; thank you very much! I appreciate your explanation.

Sorry the delay on answering but our answer was about the same.

There are 9 possibilities left for the second spot because we already used 1 digit for the first spot. ( No digit is to repeat) and also remember that we have 10 digits together from 0 to 9

 

[palmtreesarecool]

Sorry the delay on answering but our answer was about the same.

There are 9 possibilities left for the second spot because we already used 1 digit for the first spot. ( No digit is to repeat) and also remember that we have 10 digits together from 0 to 9

Awesome! Thank you very much! I appreciate that.