Okay, let's see. I need to find the value of 5 star 1 using the given operation defined as a star b equals 9a plus 2b minus ab plus 5. Hmm, so the operation is a binary operation here, meaning it takes two numbers and combines them according to that formula. Let me break it down step by step.

First, the formula is 9a + 2b - ab + 5. So, for any two numbers a and b, I substitute them into this expression. The question is asking for 5 star 1, so a is 5 and b is 1. Let me write that down.

Substituting a = 5 and b = 1 into the formula:

9*(5) + 2*(1) - (5)*(1) + 5.

Now, let's compute each term one by one to avoid mistakes. Let's start with 9 times 5. 9*5 is 45. Okay, that's straightforward.

Next term is 2 times 1. That's 2*1, which is 2. Easy enough.

Third term is -ab, which is -5*1. So that's -5. Wait, is that correct? Yes, because the formula has -ab, so substituting a=5 and b=1 gives -5*1 = -5. Got it.

Last term is +5. So that's just 5.

Now, adding all these together: 45 + 2 -5 +5. Let me compute that step by step.

First, 45 + 2 is 47. Then, 47 -5 is 42. Then, 42 +5 is 47. So the total is 47. Hmm, that seems straightforward.

Wait, let me check again to make sure I didn't make any arithmetic errors. Let's recalculate each term:

9*5 = 45. Correct.

2*1 = 2. Correct.

-5*1 = -5. Correct.

+5. Correct.

Adding them up: 45 +2 is 47. Then 47 -5 is 42. Then 42 +5 is 47. Yep, that's right. So the result is 47.

But just to be thorough, let me verify each step once more. Sometimes when doing multiple operations, it's easy to mix up signs or miscalculate.

First term: 9*5. 9*5 is indeed 45. Second term: 2*1 is 2. Third term: -5*1 is -5. Fourth term: +5. So adding them: 45 +2 is 47. Subtract 5 gives 42, then add 5 gives 47. All steps check out.

Alternatively, maybe there's a different way to approach this. Let's see. Maybe factor out some terms or see if there's a pattern, but since the operation is defined as a specific formula, substitution seems the most straightforward method here. There's no ambiguity in the operation's definition, so substitution is the way to go.

Another way to check is to substitute the values into the original equation again. Let's write the equation with a=5 and b=1:

9*5 + 2*1 - (5*1) +5.

Calculating each term:

9*5 = 45

2*1 = 2

5*1 =5, so subtracting that gives -5

Plus 5.

Adding all together: 45 +2 =47, 47 -5=42, 42 +5=47. Yep, same result. So that's consistent.

I don't see any other possible mistakes here. The arithmetic seems straightforward, and the substitution is correct. So I think the answer is 47.

Wait, just to make sure, let me consider if there's any alternative interpretation of the original operation. For example, could the formula be misread? The original problem states a star b = 9a + 2b - ab +5. So the formula is 9 times a, plus 2 times b, minus a times b, plus 5. So the order is correct. So substituting a=5 and b=1 gives 9*5 +2*1 -5*1 +5. No, that's exactly what I did. So no issues there.

Alternatively, maybe the problem was written in a different way, but according to the given, it's 9a +2b -ab +5. So no, that's the correct substitution.

Another check: maybe using order of operations. Let's ensure that multiplication is done before addition and subtraction. For example, in the term -ab, that's -(a*b), right? So in the formula, the term is -ab, which is equivalent to -(a*b). So when a=5 and b=1, that's -(5*1) = -5. Which is what I did. So correct.

So yes, all steps are correct. Therefore, the value of 5 star 1 is 47.
Therefore, the value of $5 \star 1$ is $\boxed{47}$.
</think>