Question

For \[r = 0,1,2,..........,10\] . let \[{A_r}\] , \[{B_r}\] and \[{C_r}\] denote respectively the coefficient of \[{x^r}\] in the expansion of \[{\left( {1 + x} \right)^{10}}\] , \[{\left( {1 + x} \right)^{20}}\] and \[{\left( {1 + x} \right)^{30}}\] Then \[\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} \] is

Answer 1

Hint:In order to solve the given problem and find the value of given summation of the terms. First find out the general terms for the coefficient of each series expansion by the formula of general term of expansion further simplify the summation by the use of formula for the combination terms to bring it in simplified form.

Complete step-by-step answer:
Given that:
\[r = 0,1,2,..........,10\]
And the terms
\[{A_r}\] , \[{B_r}\] and \[{C_r}\] denote denote respectively the coefficient of \[{x^r}\]
And the series for the expansion are
\[{\left( {1 + x} \right)^{10}}\] , \[{\left( {1 + x} \right)^{20}}\] and \[{\left( {1 + x} \right)^{30}}\]
As we know that for some general series of the form
 \[{\left( {1 + p} \right)^q}\]
The coefficient of general r term is given as:
\[{}^q{C_r}\]
Using the above theorem let us find the general term r of each of the given series.
\[{A_r}\] the coefficient of \[{x^r}\] in the \[{\left( {1 + x} \right)^{10}} = {}^{10}{C_r}\]
\[{B_r}\] the coefficient of \[{x^r}\] in the \[{\left( {1 + x} \right)^{20}} = {}^{20}{C_r}\]
\[{C_r}\] the coefficient of \[{x^r}\] in the \[{\left( {1 + x} \right)^{30}} = {}^{30}{C_r}\]
Now we have to find the value of the summation of the terms:
\[\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} \]
Let us now proceed further by simplifying the terms.
\[
  \sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} \\
   = \sum\limits_{r = 1}^{10} {{A_r}{B_{10}}{B_r} - \sum\limits_{r = 1}^{10} {{A_r}{C_{10}}{A_r}} } \\
 \]
Let us substitute the values found above in the given summation. We get:
\[ = \left[ {\sum\limits_{r = 1}^{10} {{}^{20}{B_{10}} \times {}^{10}{C_r} \times {}^{20}{C_r}} } \right] - \left[ {\sum\limits_{r = 1}^{10} {{}^{30}{C_{10}} \times {}^{10}{C_r} \times {}^{10}{C_r}} } \right]\]
Let us take the constant terms out of the summation.
\[ = \left[ {{}^{20}{B_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_r} \times {}^{20}{C_r}} } \right] - \left[ {{}^{30}{C_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_r} \times {}^{10}{C_r}} } \right]\]
As we know that the coefficient of combination term can be changed by the use of the following formula.
\[{}^x{C_y} = {}^x{C_{x - y}}\]
Let us use the above formula to simplify the given equation. We have:
\[ = \left[ {{}^{20}{B_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_{10 - r}} \times {}^{20}{C_r}} } \right] - \left[ {{}^{30}{C_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_{10 - r}} \times {}^{10}{C_r}} } \right]\]
As we know that the sum of combination term is given as:
\[\sum\limits_{t = 1}^n {{}^n{C_{n - t}} \times {}^m{C_t}} = \left( {{}^{n + m}{C_n} - 1} \right)\]
Let us use the above formula for summation in the above equation.
\[
   = \left[ {{}^{20}{B_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_{10 - r}} \times {}^{20}{C_r}} } \right] - \left[ {{}^{30}{C_{10}} \times \sum\limits_{r = 1}^{10} {{}^{10}{C_{10 - r}} \times {}^{10}{C_r}} } \right] \\
   = \left[ {{}^{20}{B_{10}} \times \left( {{}^{30}{C_{10}} - 1} \right)} \right] - \left[ {{}^{30}{C_{10}} \times \left( {{}^{20}{C_{10}} - 1} \right)} \right] \\
 \]
Let us now simplify the term to get the final answer.
\[
   = \left[ {\left( {{}^{20}{B_{10}} \times {}^{30}{C_{10}}} \right) - {}^{20}{B_{10}}} \right] - \left[ {\left( {{}^{30}{C_{10}} \times {}^{20}{C_{10}}} \right) - {}^{30}{C_{10}}} \right] \\
   = \left( {{}^{20}{B_{10}} \times {}^{30}{C_{10}}} \right) - {}^{20}{B_{10}} - \left( {{}^{30}{C_{10}} \times {}^{20}{C_{10}}} \right) + {}^{30}{C_{10}} \\
   = {}^{30}{C_{10}} - {}^{20}{B_{10}} \\
   = {C_{10}} - {B_{10}} \\
 \]
Hence, the final value of the term is \[{C_{10}} - {B_{10}}\]

Note: In order to solve these types of problems related to series expansion students must remember the formula for the general term of the series. Also students must remember the formula for modification of combination terms of the series and students must simplify the terms to get the final answer.