The Commutative property: Using parentheses

According to the rules of the order of operations, we first solve the exercise within parentheses from left to right:

$32-15=17$

$17-12=5$

Now, we obtain the exercise:

$5\times34\times12=$

According to the rules of the order of operations, since this is an exercise with only a multiplication operation, we rearrange the numbers to make the solution easier for us:

$5\times12\times34=$

We solve the exercise from left to right:

$5\times12=60$

$60\times34=2040$

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all else.

Therefore we'll start by simplifying the expressions in parentheses first:
$(7+2+3)(7+6)(12-3-4)=\\ 12\cdot13\cdot5$

Now we'll calculate the multiplication result step by step from left to right:

$12\cdot13\cdot5 =\\ 156\cdot5 =\\ 780$

Note that since the commutative property of multiplication applies to the expression we simplified above between all terms, the order of operations in this calculation doesn't matter (it's not necessary to perform the left multiplication first etc. as we did). However it is recommended to practice performing operations from left to right since this is the natural order of arithmetic operations (in the absence of parentheses, or other preceding arithmetic operations according to the order of operations mentioned at the beginning of this solution).

Therefore the correct answer is answer C.

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

Therefore, we'll start by simplifying the expressions in parentheses first:
$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{2}{4}\cdot10:7:5 = \\ \frac{1}{2}\cdot10:7:5$

We calculated the expression inside the parentheses by adding the fractions, which we did by creating one fraction using the common denominator (4) which in this case is the denominator in all fractions, so we only added/subtracted the numerators (according to the fraction sign), then we reduced the resulting fraction,

We'll continue and note that between multiplication and division operations there is no defined precedence for either operation, therefore we'll calculate the result of the expression obtained in the last step step by step from left to right (which is the regular order in arithmetic operations), meaning we'll first perform the multiplication operation, which is the first from the left, and then we'll perform the division operation that comes after it, and so on:

$\frac{1}{2}\cdot10:7:5 =\\ \frac{1\cdot10}{2}:7:5 =\\ \frac{10}{2}:7:5 =\\ 5:7:5 =\\ \frac{5}{7}:5$

In the first step, we performed the multiplication of the fraction by the whole number, remembering that multiplying by a fraction means multiplying by the fraction's numerator, then we simplified the resulting fraction and reduced it (effectively performing the division operation that results from it), in the final step we wrote the division operation as a simple fraction, since this division operation yields a non-whole result,

We'll continue and to perform the final division operation, we'll remember that dividing by a number is the same as multiplying by its reciprocal, and therefore we'll replace the division operation with multiplication by the reciprocal:

$\frac{5}{7}:5 =\\ \frac{5}{7}\cdot\frac{1}{5}$

In this case we preferred to multiply by the reciprocal because the divisor in the expression is a fraction and it's more convenient to perform multiplication between fractions,

We will then perform the multiplication between the fractions we got in the last step, while remembering that multiplication between fractions is performed by multiplying numerator by numerator and denominator by denominator while maintaining the fraction line, then we'll simplify the resulting expression by reducing it:

$\frac{5}{7}\cdot\frac{1}{5} =\\ \frac{5\cdot1}{7\cdot5}=\\ \frac{5}{35}=\\ \frac{1}{7}$

Let's summarize the solution steps, we got that:

$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{1}{2}\cdot10:7:5 =\\ 5:7:5 =\\ \frac{5}{7}\cdot\frac{1}{5} =\\ \frac{1}{7}$

Therefore the correct answer is answer B.

According to the order of operations, we will first solve the expression in parentheses.

Note that since the denominators are not common, we will look for a number that is both divisible by 2 and 3. That is 6.

We will multiply one-third by 2 and one-half by 3, now we will get the expression:

$\frac{1}{4}\times(\frac{2+3}{6})=$

Let's solve the numerator of the fraction:

$\frac{1}{4}\times\frac{5}{6}=$

We will combine the fractions into a multiplication expression:

$\frac{1\times5}{4\times6}=\frac{5}{24}$