How do you use binomial theorem to expand and simplify the expression $ {\left( {x + 1} \right){^6}} $ ?

Hint: In order to find the expansion of the expression $ {\left( {x + 1} \right){^6}} $ using binomial theorem, we will use the formula from the binomial theorem, by which we can expand up to $ {n^{th}} $ polynomial. Then, by substituting the values and evaluating it, we will determine the required expansion.

Complete step-by-step answer:
The binomial theorem describes the algebraic expansion of powers of a binomial. According to this theorem, we are able to expand the polynomial $ {\left( {x + y} \right)^n} $ into a sum involving terms of the form $ a{x^b}{y^c} $ , where the exponents $ b $ and $ c $ are non-negative integers with $ b + c = n $ , and the coefficient $ a $ of each term is a specific positive integer depending on $ n $ and $ b $ .
 $ {\left( {x + y} \right)^n} = \sum\limits_{r = 0}^n {\left( {\begin{array}{*{20}{c}}
  n \\
  r
\end{array}} \right){x^{n - r}}{y^r}} $
Now, the given expression is $ {\left( {x + 1} \right){^6}} $ . Thus, $ x = x $ , $ y = 1 $ and $ n = 6 $ .
Therefore, let us substitute the values, we have,
 $ {\left( {x + 1} \right){^6}} = {^6}{C_0}\left( {{x{^6}}} \right)\left( {{1^0}} \right) + {^6}{C_1}\left( {{x^5}} \right)\left( {{1^1}} \right) + {^6}{C_2}\left( {{x^4}} \right)\left( {{1^2}} \right) + {^6}{C_3}\left( {{x^3}} \right)\left( {{1^3}} \right) + {^6}{C_4}\left( {{x^2}} \right)\left( {{1^4}} \right) + {^6}{C_5}\left( {{x^1}} \right)\left( {{1^5}} \right) + {^6}{C_6}\left( {{x^0}} \right)\left( {{1{^6}}} \right) $ where, $ n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $
Therefore, $ {^6}{C_1} = 0 $ , $ {^6}{C_1} = 6 $ , $ {^6}{C_2} = 15 $ , $ {^6}{C_3} = 20 $ , $ {^6}{C_4} = 15 $ , $ {^6}{C_5} = 6 $ and $ {^6}{C_6} = 1 $ .
 $ {\left( {x + 1} \right){^6}} = 1\left( {{x{^6}}} \right)\left( 1 \right) + 6\left( {{x^5}} \right)\left( 1 \right) + 15\left( {{x^4}} \right)\left( 1 \right) + 20\left( {{x^3}} \right)\left( 1 \right) + 15\left( {{x^2}} \right)\left( 1 \right) + 6\left( x \right)\left( 1 \right) + 1\left( 1 \right)\left( 1 \right) $
 $ {\left( {x + 1} \right){^6}} = {x{^6}} + 6{x^5} + 15{x^4} + 20{x^3} + 15{x^2} + 6x + 1 $
Hence, the expansion $ {\left( {x + 1} \right){^6}} $ of the expression using binomial theorem is, $ {x{^6}} + 6{x^5} + 15{x^4} + 20{x^3} + 15{x^2} + 6x + 1 $ .
So, the correct answer is “ $ {x{^6}} + 6{x^5} + 15{x^4} + 20{x^3} + 15{x^2} + 6x + 1 $ ”.

Note: The binomial theorem is also known as binomial expansion. As the power increases the expansion becomes lengthy and tedious to calculate. A binomial expression that has been raised to a very large power can be easily calculated with the help of Binomial theorem. To find binomial coefficients we can use Pascal’s triangle also.
Binomial theorem is heavily used in probability theory, and a very large part of the US economy depends on probabilistic analyses. It is most useful in our economy to find the chances of profit and loss which is a great deal with developing economies. Moreover binomial theorem is used in forecasting services. The future weather forecasting is impossible without a binomial theorem. The disaster forecast also depends upon binomial theorem.