Simplify the following: $\left( {\dfrac{3}{{11}} \times \dfrac{5}{6}} \right) - \left( {\dfrac{9}{{12}} \times \dfrac{4}{3}} \right) + \left( {\dfrac{5}{{13}} \times \dfrac{6}{{15}}} \right)$

Hint: According to given in the question we have to find the value of expression so, first of all to solve the expression we have to follow the BODMAS rule as explained below:
B – Brackets of (according to this first of all we have solve the smaller bracket then we have to solve curly bracket and after it we have to solve large bracket)
O- Order
D – Divide
M – Multiply
A – Subtract
A – Addition
So, with the help of BODMAS rule first of all we will solve the terms inside the smaller bracket and after that we will add then subtract the terms obtained.

Complete step-by-step answer:
Given expression: $\left( {\dfrac{3}{{11}} \times \dfrac{5}{6}} \right) - \left( {\dfrac{9}{{12}} \times \dfrac{4}{3}} \right) + \left( {\dfrac{5}{{13}} \times \dfrac{6}{{15}}} \right)$…………………..(1)
Step 1: To solve the expression first of all we will simplify the term $\left( {\dfrac{3}{{11}} \times \dfrac{5}{6}} \right)$
Hence,
$ = \dfrac{{15}}{{66}}$
Now dividing with 3, to the numerator and denominator of the obtained fraction.
$ = \dfrac{5}{{22}}$
Step 2: Same as the step 1 we will simplify the term $\left( {\dfrac{9}{{12}} \times \dfrac{4}{3}} \right)$of the given expression.
Hence,
$
   = \dfrac{{36}}{{36}} \\
   = 1 \\
 $
Step 3: Now, same as the step 1 and step 2 we have to simplify the term $\left( {\dfrac{5}{{13}} \times \dfrac{6}{{15}}} \right)$given in the expression.
Hence,
$ = \dfrac{{30}}{{195}}$
On solving the fraction obtained just above,
$ = \dfrac{2}{{13}}$
Step 4: Now, we have to substitute all the simplified terms in the expression (1).
$ = \dfrac{5}{{22}} - 1 + \dfrac{2}{{13}}$
Hence, to simplify the obtained expression we have to find the L.C.M.
$
   = \dfrac{{65 - 286 + 44}}{{286}} \\
   = \dfrac{{109 - 286}}{{286}} \\
   = - \dfrac{{177}}{{286}} \\
 $

Hence, with the help of BODMAS rule we have obtained the value of the given expression: $\left( {\dfrac{3}{{11}} \times \dfrac{5}{6}} \right) - \left( {\dfrac{9}{{12}} \times \dfrac{4}{3}} \right) + \left( {\dfrac{5}{{13}} \times \dfrac{6}{{15}}} \right)$$ = - \dfrac{{177}}{{286}}$

Note: In the BODMAS rule O belongs to order that means power and roots but before that we have to solve the terms inside the brackets first smaller after that curly and after that large bracket.
The order of operation is a set of standard rules that reflect convention about which process to perform to evaluate as a given numeric expression.