Question

Find the value of given expression,
\[\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}} \right)+..........+\left( {}^{7}{{C}_{6}}+{}^{7}{{C}_{7}} \right)\]
a.\[{{2}^{8}}-2\]
b.\[{{2}^{8}}-1\]
c.\[{{2}^{8}}+1\]
d.\[{{2}^{8}}\]

Answer 1

Hint:Separate the terms present in the bracket and arrange them to form a combination. Do the required adjustments and use the formula \[\left( {}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+...+{}^{n}{{C}_{n}} \right)={{2}^{n}}\] to get the final answer.

Complete step by step answer:
To solve the given expression we will write it down first and assume it as ‘S’, therefore we will get,
\[\therefore S=\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}} \right)+..........+\left( {}^{7}{{C}_{6}}+{}^{7}{{C}_{7}} \right)\] ……………………………………………. (1)
If we separate the first and second terms of the bracket then they will form different series as shown below,
\[\therefore S=\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}}+...+{}^{7}{{C}_{6}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}}+....+{}^{7}{{C}_{7}} \right)\]
If we see the above equation carefully then we can say that in the first bracket the term \[{}^{7}{{C}_{7}}\] is missing to complete the combination and in second bracket the term \[{}^{7}{{C}_{0}}\] is missing to complete the combination therefore we will add and subtract \[{}^{7}{{C}_{0}}+{}^{7}{{C}_{7}}\] in the above equation therefore we will get,
\[\therefore S=\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}}+...+{}^{7}{{C}_{6}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}}+....+{}^{7}{{C}_{7}} \right)+{}^{7}{{C}_{0}}+{}^{7}{{C}_{7}}-{}^{7}{{C}_{0}}-{}^{7}{{C}_{7}}\]
Now, if we arrange the added terms with particular brackets to cover the combination we will get,
\[\therefore S=\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}}+...+{}^{7}{{C}_{6}}+{}^{7}{{C}_{7}} \right)+\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}}+{}^{7}{{C}_{2}}+....+{}^{7}{{C}_{7}} \right)-{}^{7}{{C}_{0}}-{}^{7}{{C}_{7}}\]
To proceed further in the solution we should know the formula given below,
Formula:
\[\left( {}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+...+{}^{n}{{C}_{n}} \right)={{2}^{n}}\]
By using the above formula in ‘S’ we will get,
\[\therefore S={{2}^{7}}+{{2}^{7}}-{}^{7}{{C}_{0}}-{}^{7}{{C}_{7}}\]
As we know that the value of \[{}^{7}{{C}_{0}}\] and \[{}^{7}{{C}_{7}}\] is 1 and if we put this value in the above equation we will get,
\[\therefore S={{2}^{7}}+{{2}^{7}}-1-1\]
By doing addition in the above equation we will get,
\[\therefore S=2\times {{2}^{7}}-2\]
Above equation can also be written as,
\[\therefore S={{2}^{1}}\times {{2}^{7}}-2\]
To proceed further in the solution we should know the formula given below,
Formula:
\[{{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}\]
By using the above formula in ‘S’ we will get,
\[\therefore S={{2}^{1+7}}-2\]
By simplifying the above equation we will get,
\[\therefore S={{2}^{8}}-2\]
If we compare the equation with equation (1) we will get,
\[\therefore \left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}} \right)+..........+\left( {}^{7}{{C}_{6}}+{}^{7}{{C}_{7}} \right)={{2}^{8}}-2\]
Therefore the value of the expression \[\left( {}^{7}{{C}_{0}}+{}^{7}{{C}_{1}} \right)+\left( {}^{7}{{C}_{1}}+{}^{7}{{C}_{2}} \right)+..........+\left( {}^{7}{{C}_{6}}+{}^{7}{{C}_{7}} \right)\] is equal to \[{{2}^{8}}-2\].
Therefore the correct answer is option (a).

Note: Do not use the formula \[{}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!\times r!}\] as it will complicate your solution and probably you will not get the answer in the required format.