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.