A set of license plates has three letters followed by 3...

A set of license plates has three letters followed by 3 digits. (a) In how many ways can a license plate be made with three letters followed by 3 digits? (Letters or digits may be repeated.) (b) In how many ways can a license plate be made with three different letters followed by 3 different digits? (c) If a license plate is made with three letters followed by 3 digits, what is the probability that it has no repeated letters or digits? (Enter your probability as a fraction.)

Transcript

00:01 Part a, we have three letters followed by three digits, and the letters or digits can be repeated.

00:07 We wanna know the total number of license plates there could be.

00:09 So there's 26 letters in the alphabet, and then there's 10 digits to choose from between zero and nine.

00:16 So we get, since it can be repeated, we get 26 times 26 times 26 times 10 times 10 times 10.

00:26 You could write this as 26 to the third times 10 to the third, and 26 to the third is 17 ,576 times 10 to the third, which is 1 ,000.

00:48 17 ,576 times 1 ,000, i get 17 ,576 ,000.

01:03 Part b, we wanna know the total number of license plates that can be made with three different letters followed by three different digits.

01:12 So the first time i have 26 letters to choose from.

01:16 Then the next time there's 25 and then 24.

01:20 And then for the numbers, there's 10 numbers the first time, then there's nine numbers, and then there's eight because we don't wanna repeat them.

01:29 So 26 times 25 times 24 is 15 ,600 times 10 times nine times eight is 720.

01:40 15 ,600 times 720 is 11 ,232 ,000.

01:54 Part c, we wanna know the probability that it has no repeated letters or digits.

02:03 So the first time, remember there's 26 letters to choose from.

02:07 So we're picking one letter out of the 26, but we have all 26 letters to choose from.

02:17 And then the next time i can choose from 25 letters out of the 26.

02:22 And then i can choose from 24 letters out of the 26.

02:26 And then for the numbers, the first time i can choose any of the 10 numbers out of the 10 numbers.

02:31 The second time, since we don't wanna repeat, i have nine numbers to choose from out of 10 numbers.

02:39 And then the next time i have eight numbers to choose from out of 10 numbers.

02:43 And with fractions, we multiply straight across.

02:46 So on the top, i have 26 times 25 times 24 times 10 times nine times eight, 11 ,232 ,000 out of, and then we have 26 times 26 times 26 times 10 times 10 times 10...

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