I write a two-digit number by randomly picking each digit to be either three or seven, where digits can be repeated. Write the set of all possible outcomes.
Well this is a case of writing any two-digit number you like, so long as all of the digits are either three or seven. Now it pays to try and be a bit methodical about this process. Rather than just randomly writing down combinations of three or seven, let’s do this in a sort of an ordered way.
I’m trying to come up with two-digit numbers, and I can only choose from three and seven when I’m picking those digits. So let’s start off with thinking about the three. What if I put three in both boxes, so that’s three for the first digit, three for the second digit. The number then would be thirty-three. Well another option would be to choose three first still, but now choose seven as the other digit. So now we’ve explored both possibilities that we can get, if we have three as our first digit. So we can either have three as the second digit or seven as the second digit.
Now let’s explore the possibilities of having seven as the first digit. Well we could either have three as the second digit, or we could have seven as the second digit.
Now if we look at the list of numbers that we’ve come up with, you can see some patterns here. So if we look at the second digit there, there’s a little pattern that goes, three, seven, three, seven. And if we look at the first digit, we’ve got all the possibilities for three as the first digit, followed by all the possibilities for seven as the first digit. Now this methodical way of organising our analysis, means we much less likely to miss out any possibilities, or to double count possibilities.
Now one last thing, the question asked us to write the set of all possible outcomes. So we need to write this in set notation. And there we have in set notation. Now if you’ve written your numbers in a slightly different order within that set, that doesn’t matter. It’s still equivalent. So long as you’re methodical about how you do it, you’re less likely to make a mistake.