Brackets/parentheses (depends where you live)
Exponents/orders (depends)
Division
Multiplication
Addition
Subtraction

I can see the point of having brackets, since then you can change the order, and exponents net, but why did they put div/multiply before addition and subtraction? Is there an advantage to that? Or was it just randomly chosen that way?

That multiplication comes before addition is a convention—that is, it was chosen that way. But not randomly. It makes mathematical expressions easier and less awkward to work with, because of properties of addition and multiplication that aren’t just convention, such as the fact that multiplication distributes over addition.

Pre-modern usage about order of operations may have been influenced by very ancient features of the construction of number words. We say, for example, “four hundred and twenty-nine”, i.e., four hundreds and two tens and nine units. We don’t add a hundred and a ten and a unit together and then multiply the sum by something.

So the “multiplication before addition” convention may have roots in elementary aspects of combining groupings of different measures.

Ah, that’s insightful! Now that you mention it, I can see the parallel between an expression like “There are three sevens and four nines” (i.e. 3\cdot7 + 4\cdot9) and an expression like “There are three cats and four dogs,” or even “There are black cats and brown dogs.”

In some kinds of real-life problems, “sum of products” comes up. That is, you have a bunch of pairs of numbers, you multiply the two numbers in each pair, then add up those products. It turns out, this pattern of operations turns up in immensely many contexts. On the contrary, I don’t think “product of sums” comes up nearly so often.

As mentioned above, this order of operations is just a standardized convention, decided upon because it is useful. There is no inherent mathematical reason for it. Note also, the need to specify an order of operations is necessary, to begin with, because of our traditional in-fix notation for arithmetic expressions. There are other notations for arithmetic – notably Polish (prefix) notation and Reverse Polish (post-fix) notation, in which the operations are simply done in the order they are written from left to right. No BEDLAM at all with that!

Here are two examples of common real-life problems that involve a sum of products:

(1) Adding up your grocery bill:
You buy 6 cans of beans for 1.38 each; a dozen lemons for 0.30 each; and 2 bottles of milk for 3.89 each. How much does this cost?

Solution:

You multiply the amount of each item by the quantity of each item (these are the products), then add those up (sum of those products):
( 6 * 1.38 ) + ( 12 * 0.30 ) + ( 2 * 3.89 ) = 19.66
Note that the parentheses in the above are redundant because of the BEDMAS rules, but are always good to include to make it explicitly clear. This is the kind of calculation for which those BEDMAS rules were made!

(2) That calculation near and dear to the hearts of all secondary school and college students: Your grade-point average.

General Procedure

Multiply your grade in each class (a number from 0 to 4 in most common systems) by the number of units in that class – These are the products, which are the number of grade points for each class.

Then add those up to get the total grade points (this is the sum of those products).

Then also add up the total units of all your classes, to get your total unit load. Then divide the total grade points by that, and there’s your grade-point average. Again note the “sum of products” in this calculation.

Here’s another example you are all familiar with even if you don’t know it: Finding the center of gravity of your loaded airplane. This is critical: The center of gravity must lie within a very narrow band or your airplane won’t get off the ground (or worse still, it will get off the ground but be very unstable in the air). There are worksheets where you fill in the positions of all your passengers and their weights, the position and weight of any baggage, the amount and weight of the fuel on board, and maybe even the amount and weight of the engine oil. Then there is a specific sequence of calculations to compute the location of the center of gravity. And guess what: It involves a sum of products!

And how is it that you already know this procedure even if you don’t know you know it? Because the steps are exactly the same, step for step, as calculating your grade-point average! Both of these calculations are just examples of the weighted average, which comes up in a whole lot of situations.

In most everyday situation, order is fairly obvious from context. If you have a list of item quantities and item prices you multiply the quantity by the price and then sum. If you have a list of side lengths of your fence and you’re need three boards per side, you add up the lengths and then multiply by two. You probably don’t bother with a parenthesis though, you just write a large X2 and you know what you meant.

I think the convention comes from equations. We have a lot of expression of the type 2x + 3y and 2(3x + y). With the reversed order of operations they would have been (2x) + (3y) and 2(3x) + y. And with no order of operations (2x) + (3y) and 2((3x) + y). There would have been a lot more parentheses.

That’s ignoring implied multiplication, the idea that 2x is understood as 2*x. We could have had addition/subtraction before multiplication, but implied multiplication before either.

All of which illustrates why RPN is easier and why I find an RPN calculator easier to use. Unambiguously, 3,4,5+\times=27, while 3,4,5\times+=60. The commas represent the enter key on the calculator. No parens ever needed,

Another point is that anyone who writes 3+4\times 5 is asking to be misunderstood. Any mathematician would use parens.

Hmm, I always thought it was because usually, multiplication (over the natural numbers) is defined recursively in terms of addition. So x\cdot S(y), where S() is the successor function, is given by x+x\cdot y, which together with x\cdot 0 = 0 then allows one to uniquely resolve any multiplication in terms of repeat addition. So in that sense, 3+4\cdot 5 can be reduced to 3 + 4 + 4 + 4 + 4 + 4.

Although one can suppose that this just kicks the can down the road to the question of why multiplication is defined in this way. But it’s always made sense to me that you’re basically just resolving everything in terms of the more ‘basic’ operation.

Works for higher-order operations, too: exponentiation is defined recursively in terms of multiplication, so you resolve x^y into \underbrace{x\cdot x \cdot \ldots \cdot x}_{\text{y times}}, which can then be further resolved into sums.

The possible problem with RPN: I’ve thought about whether the whole body of algebra could be re-written using RPN instead of the traditional notation, and I decided that wouldn’t work as well.

One particular problem is in the notation for division: Traditional notation uses a numerator on top, with a horizontal line below that, and the denominator below that. This makes it fairly easy to spot common factors in the numerator and denominator, which can then be “canceled”. This is harder to see if the whole thing is written using RPN.

You may or may not write algebra using Polish notation, but when you’re doing the calculations, you almost always do. It may look like you’re using other methods, but those mostly end up converting to Polish. Which is why just using Polish in the first place is more efficient: No conversions needed.