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# Help Explaining Probabilities

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The license plates in Eulerville are made of seven distinct characters, six of which are letters from A to Z inclusive and one of which is a digit from 0 to 9 inclusive. How many different Eulerville license plates include
the word “MATH” with those four letters consecutive and in that order?

I just need to figure out how to get it, because my way didn't seem correct.

I did 26 choose 6, since there are letters and the numbers aren't necessary to be counted, and when I computed it, I got: 26x25x24x23x22x21 choose 6!, which came out as 6630624/720, simplifying to 9209.2. I dont believe my answer is correct, given that I had a decimal point in my answer. Could anyone try to help me understand how to solve this?

Dec 27, 2020

#1
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MATH     can sit in 4 positions

then there are 3 positions left , one has to be a number     10 choices

the other two positions can be a letter    26 x 26

but the number can be in one of three positions     x 3

4  x 10  x 26 x 26 x 3      = 81120   (?)

Dec 27, 2020
#2
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Ok, so I checked the answer and it was 55,440, although I still don't know how to get it.

Guest Dec 27, 2020
#5
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Cphill answer below..... I neglected the 'distinct' portion of the question

so it should be :

MATH     can sit in 4 positions

then there are 3 positions left , one has to be a number     10 choices

the other two positions can be a letter    22 x 21     (since   M   A   T   H  are already used)

but the number can be in one of three positions     x 3

4  x 10  x 22 x 21  x 3      = 55440

Guest 21 mins ago

Guest Dec 27, 2020
#3
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Note....for any plate..... the word "MATH" can appear in  any one of 4 positions

Since  the  characters are distinct, I'm assuming that  no letter can be repeated

In the other 3 positions, we  can choose  any of  22 remaining letters * any of 21 remaining letters * 10 digits  and these can be arranged in 3!  = 6 ways

So.....in this scenario, we have

4 * 22 * 21 * 10 * 3    =

55440 possible plates

Dec 27, 2020
#4
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Thank you CPhill! It really helps out a lot!

Guest Dec 27, 2020