Answer:
[tex]((3 * 7) - 6) * 5[/tex]
Step-by-step explanation:
For this problem, we will use trial and error.
It will take a while, but we can reduce the amount of guesses we have to make.
Looking at the problem, it is obvious that we will require multiplication at some point, as adding all the numbers together is nowhere near 75.
Let's start this problem by finding the prime factors of 75:
75 = [tex]25 * 3[/tex]
75 = [tex]5 * 5 * 3[/tex]
The prime factors of 75 are: [tex]5 * 5 * 3[/tex].
Our numbers are 7, 5, 6, and 3.
We already have a 5 and a 3, so we need to use 7 and 6 to get 5, and that's impossible, so this is a pass.
Next, we'll look at another factor, [tex]25 * 3[/tex].
We already have a 3, so we can ignore that. We need to use 7, 6, and 5 to get 25.
Here are some of the results:
[tex](5 * 6) - 7 = 23[/tex] (Nope)
[tex](7 * 5) - 6 = 29[/tex] (Nope)
[tex]6 + 5 + 7 = 18[/tex] (Nope)
There's more combinations, but the numbers are either too large or too small. This is a pass too.
Now, we'll try another factor, [tex]15 * 5[/tex].
We already have a 5, so let's ignore it and use 7, 6, and 3 to get 15.
[tex]7 + 6 + 3 = 16[/tex] (Nope)
[tex](6 * 3) - 7 = 11[/tex] (Nope)
[tex](3 * 7) - 6 = 15[/tex] (Yep)
So our expression to make 15 is [tex](3 * 7) - 6[/tex]. Then we multiply the whole thing by 5. Thus our expression is:
[tex]((3 * 7) - 6) * 5[/tex]
Let's check our answer:
[tex]((3 * 7) - 6) * 5[/tex]
[tex](21 - 6) * 5[/tex]
[tex]15 * 5[/tex]
[tex]75[/tex]
The expression gives us 75, which is what we want, so our answer is correct.
