A number lock on a suitcase has 3 wheels each labelled with...

A number lock on a suitcase has 3 wheels each labelled with 10 digits from 0 to 9 . If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible?
(a) 720
(b) 760
(c) 680
(d) 780

Key Concepts

Permutations

Permutations refer to the arrangements of a set of items where the order matters and no item is repeated. In problems like this, the focus is on counting the total number of distinct sequences that can be formed from a given set of items without repetition.

Fundamental Counting Principle

The Fundamental Counting Principle is a method to determine the total number of outcomes for a sequence of events. It involves multiplying the number of choices available at each step, which is particularly useful when calculating the total number of ways to form an ordered sequence from a set of choices.

Transcript

00:01 We are told that we are dealing with a three -number combination lock on a suitcase, and we can possibly have the digits zero to nine on any given one of those wheels, any given one of those spots.

00:13 So then we are asked how many possible combinations there could be of those three numbers, and first it says assuming that no digit is repeated.

00:23 So if we assume that no digit is repeated in thinking about this, we have three -digit combination.

00:29 So for the first digit we have all the possible digits pop.

00:34 For the first number we have all the possible digits.

00:36 So that'd be 0 through 9.

00:39 0 1, 2, 3, 4, 5, 6, 7, 8, 9.

00:46 Well, if you count all those up, those are, that is 10 possible digits that we have.

00:53 But if we think about the next number in our combination, well there's only going to be 9 possible digits here because depending on which of these 10 that we just used there's only going to be nine left because it said we can't have any digit be repeated likewise our third spot is only going to have eight possible digits because we've already used two of our original 10 for the first two slots which means there's only eight left because we can't repeat say for example that zero was the first digit well that's why there's only nine available in the second spot because it's only 1, 2, 3, 4, 5, 6, 7, 8, 9.

01:33 And then let's say that 1 was the next number.

01:38 Well, that's why we only have 8 possible digits in the third spot because now it's only 2, 3, 4, 5, 6, 7, 8.

01:45 And so then to find our total possible combinations, we would simply need to take 10 times 9 times 8, which would give us 720 possible combinations.

02:03 Now, that is not the end of this problem, because it then asks us a separate question...

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