Answer

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**Hint:**Here in the question, we have to simplify the given expression, In simplifying an expression first of all, the bar must be removed. After removing the bar, the brackets must be removed, strictly in the order \[\left( {} \right)\], \[\left\{ {} \right\}\] and \[\left[ {} \right]\]. After removing the brackets further simplify the expression by using the BODMAS rule.

**Complete step-by-step solution:**

An arithmetic expression that involves multiple operations such as addition, subtraction, multiplication, and division are not easy to solve as compared to operations involving two numbers. An operation on two numbers is easy, but how to solve an expression with brackets and multiple operations and how to simplify a bracket?

We must remember the word BODMAS in solving sums on simplification. These letters stand for vinculum (vinculum means bar), bracket, of, division, multiplication, addition and subtraction respectively. The sums on simplification must be solved in that order i.e., first solve the vinculum followed by bracket and so on until the sum is solved.

In simplifying an expression first of all, the bar must be removed. After removing the bar, the brackets must be removed, strictly in the order \[\left( {} \right)\], \[\left\{ {} \right\}\] and \[\left[ {} \right]\]. After removing the brackets, we must use the following operations strictly in the order: (i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction

Consider the given expression

\[ \Rightarrow \,\,\,29 - \left[ {16 - 17 - 3\left( {18 - \overline {19 - 16} } \right)} \right]\]

First simplify the numbers below the bar

\[ \Rightarrow \,\,\,29 - \left[ {16 - 17 - 3\left( {18 - 3} \right)} \right]\]

Next simplify the numbers inside the \[\left( {} \right)\] bracket

\[ \Rightarrow \,\,\,29 - \left[ {16 - 17 - 3\left( {15} \right)} \right]\]

On simplify the \[\left( {} \right)\] bracket, we get

\[ \Rightarrow \,\,\,29 - \left[ {16 - 17 - 45} \right]\]

Next by the simplifying bracket order remove \[\left[ {} \right]\] bracket by simplifying its inside numbers.

\[ \Rightarrow \,\,\,29 - \left[ { - 46} \right]\]

On simplify the \[\left[ {} \right]\] bracket, we get

\[ \Rightarrow \,\,\,29 + 46\]

\[ \Rightarrow \,\,\,75\]

Hence, by The VBODMAS rule the value of \[29 - \left[ {16 - 17 - 3\left( {18 - \overline {19 - 16} } \right)} \right]\] is

**Hence the correct answer is option ‘A’.**

**Note:**The problem involves the brackets, we have a separate method to solve the problem. In brackets we have four kinds namely bar bracket, common bracket, flower bracket and the square bracket. We solve the problem involving the bracket we follow the procedure.

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