Answer:
Step-by-step explanation:
<u>Question from image</u>:
"The digits 1, 2, 3, 4, and 5 can be formed to arrange 120 different numbers. How many of these numbers will have the digits 1 and 2 in increasing order? For example, 14352 and 51234 are two such numbers."
Let's start by taking a look with the number 1. There are four possible places 1 could be, because there needs to be space for the 2 after it.
Checkmarks mark where the 1 can be, the <em>x</em> marks where it cannot be.
Let's start with the first position:
There are four places the 2 can be. For each of these four places, we can arrange the remaining 3 digits in ways. Therefore, there are
possible numbers when 1 is the first digit of the number.
Continue this process with the remaining possible positions for 1.
Second position:
There are three places the 2 can be, since the 2 must be behind the 1. For each of these three places, the remaining 3 digits can be arranged in ways. Therefore, there are
possible numbers when 1 is the second digit of the number.
The pattern continues. Next there will be 2 places to place the 2. For each of these, there are ways to rearrange the remaining 3 digits for a total of
possible numbers when 1 is the third digit of the number.
Lastly, when 1 is the fourth digit of the number, there is only 1 place the 2 can be. For this one place, there are still ways to rearrange the remaining three numbers. Therefore, there are
possible numbers when 1 is the fourth digit of the number.
Thus, there are numbers that will have the digits 1 and 2 in increasing order, from a set of 120 five-digit numbers created by the digits 1 through 5, where no digit may be repeated.
