Forming License Plate Numbers How many different license plate numbers can be made using 2 lette

step 1 All right, so we're making license plate numbers. How many different license plates can be made

Forming License Plate Numbers How many different license...

Key Concepts

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.

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.

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.

Transcript

00:01 All right, so we're making license plate numbers.

00:03 How many different license plates can be made using two letters, followed by four digits, zero through nine, and letters and digits may be repeated.

00:13 And if letters may be repeated, but digits may not be repeated, and if neither letters or digits may repeat it.

00:22 So we have three different scenarios here.

00:24 We're going to have to do three different problems.

00:27 So in this case, we're going to be looking at since we have we're going to be ordering two different kinds of things here it seems we have two letters and then we have four digits so if we're trying to order these we can't just do it in one permutation or one combination because those are meant to deal with two groups or because permutations and combinations are meant to deal with one group in this case we have letters and we have digits.

01:02 So i think the best way to go about this one is to do fundamental counting principles since we have two different types of items.

01:10 And now we have to kind of consider this scenario here.

01:14 Letters and digits may be repeated.

01:16 Let's go ahead and do that as our first scenario.

01:19 And already we know that a permutation and a combination won't work on this anyway since digits may be repeated.

01:25 Now we have two letters and we have four digits.

01:30 So we have have a total of six spots and we know that this is going to be a letter we know that this is going to be these two are going to be letters and we know that these are going to be digits so now that we know that we can go ahead and start to approach this problem and figure out exactly how many positions we have or exactly how many possibilities we have for each license plate so for our first scenario here letters and digits may be repeated.

02:05 So both can be repeated.

02:06 In this case, we have two letters in the front, and there are 26 letters in the alphabet.

02:11 So for our first two, since we can repeat, we have 26 choices and 26 choices for our first two.

02:18 Now, for our last four positions, we have four digits from zero through nine.

02:24 So that's a total of 10 choices.

02:25 So for our first digit, we can choose 10 things.

02:28 And since they may be repeated, we still have 10 choices for the rest of the spaces.

02:35 Now, by fundamental counting principle, we know that if there's one way to put this letter in here, or there's however many ways to put this letter in here, however many ways to put a letter in the second position, the total number of ways is to multiply the two.

02:49 And the best part about this question is that this can also be applied right on through to the digits, since we're just talking about one row of positions here.

03:00 So we have two letters and all of the possibilities of those two multiplied together multiplied by the number of possibilities of these four digits or the four spots with the digits in them.

03:13 So this is going to get us our final answer.

03:17 And if we go ahead and plug that into a calculator, that's going to give us a large number here...

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