Forming License Plate Numbers How many different license...

Forming License Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through $9,$ if: (a) Letters and digits may be repeated? (b) Letters may be repeated, but digits may not be repeated? (c) Neither letters nor digits may be repeated?

Forming License Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through $9,$ if:
(a) Letters and digits may be repeated?
(b) Letters may be repeated, but digits may not be repeated?
(c) Neither letters nor digits may be repeated?

Key Concepts

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics that states if one event can occur in 'm' ways and a subsequent independent event can occur in 'n' ways, then the overall process can occur in m × n ways. This principle is used to break down complex counting problems into a sequence of simpler, independent choices.

Counting with Repetition

Counting with repetition allows the same element to be chosen multiple times, meaning that each selection is independent and the number of available options remains constant at each step. This concept is often applied when the same symbols or figures can be reused in different positions of a sequence.

Counting without Repetition

Counting without repetition is a method where each item is used only once, causing the number of choices to decrease with each successive selection. This restriction leads to using permutation formulas where the order of selection matters, typically represented by a factorial expression or the formula n!/(n-k)! when selecting k items from a total of n.

Permutation

A permutation is an ordered arrangement of items where the sequence is important. In scenarios where items are not repeated, permutations are calculated by multiplying a series of decreasing positive integers, which captures the essence of ordering without reuse of elements.

Transcript

00:01 We are creating license plates with a template like i've drawn here on the left.

00:03 That is, we're going to have two letters and four numbers.

00:09 All of these are going to be randomly chosen.

00:11 So let's figure out how many different license plates we can make.

00:14 Starting off, the first condition is that we can repeat letters and we can repeat numbers.

00:19 So if this is the case, then for letters, we have 26 options for the first, 26 options for the second, and then for the numbers, there's 10 digits, 0 through 9, 10 digits for the next, 10 digits for the next, 10 digits for the next.

00:33 Well, according to the fundamental principle of counting, if we multiply together all of our number of choices, we can get our total number of different options.

00:43 So if you multiply 26 by 26 by 10 by 10 by 10, you will get a total of 6 ,760 ,000 different license plate combinations.

00:57 So if we are allowed to repeat both letters and numbers we will get 6 ,760 ,000 different plates that we can make.

01:05 All right, now let's throw some restrictions on you, make it a little bit more difficult.

01:09 So now, suppose that we can still repeat the letters, but we can't repeat the numbers anymore.

01:15 Well, now for the letters, there's still 26 times 26 different options.

01:20 However, for the numbers, so we can choose one number very easily.

01:24 Let's say we choose one.

01:26 Then for our next number, we can choose any number but one...

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