Question

Help pls I'll give 25 points

Answer 1

`\text{C. }60`

Explanation:

Question from image:

"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 x marks where it cannot be.

`\underline{\checkmark}\:\underline{\checkmark}\:\underline{\checkmark}\:\underline{\checkmark}\:\underline{X}`

Let's start with the first position:

`\underline{\checkmark}\:\underline{\#}\:\underline{\#}\:\underline{\#}\:\underline{\#}`

There are four places the 2 can be. For each of these four places, we can arrange the remaining 3 digits in
`3!=6` ways. Therefore, there are
`4\cdot 6=\boxed{24}` possible numbers when 1 is the first digit of the number.

Continue this process with the remaining possible positions for 1.

Second position:

`\underline{X}\underline{\checkmark}\:\underline{\#}\:\underline{\#}\:\underline{\#}`

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
`3!=6` ways. Therefore, there are
`3\cdot 6=\boxed{18}` 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
`3!=6` ways to rearrange the remaining 3 digits for a total of
`2\cdot 6=\boxed{12}` 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
`3!=6` ways to rearrange the remaining three numbers. Therefore, there are
`1\cdot 6=\boxed{6}` possible numbers when 1 is the fourth digit of the number.

Thus, there are
`24+18+12+6=\boxed{60}` 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.