Question

If the coefficients of  \[{{\left( 2r+e \right)}^{th}}~\] term and  \[{{\left( 3+4 \right)}^{th}}~\] term in the expansion of   \[{{\left( 1+x \right)}^{21}}~\]are equal, then find r.

Answer 1

HINT: -
The formula for writing the \[{{\left( r+1 \right)}^{th}}\] term of an expansion of \[{{(a+b)}^{n}}\] is given as follows
\[{{T}_{r+1}}={}^{n}{{C}_{r}}\left( {{a}^{n-r}} \right)\cdot \left( {{b}^{r}} \right)\]
(Where \[{}^{n}{{C}_{r}}\] is equal to \[\dfrac{n!}{r!\left( n-r \right)!}\] )
Now, for the expansion of the specific series that is \[{{\left( 1+x \right)}^{n}}~\] which is asked in the question as well, we can write the following to get the \[{{\left( r+1 \right)}^{th}}\] term of its expansion and that is
\[{{T}_{r+1}}={}^{n}{{C}_{r}}\cdot \left( {{x}^{r}} \right)\]

Complete Step-by-Step solution:
As mentioned in the question, we have to find the value of r.
Now, using the formula given in the hint to find the value of the \[{{\left( 2r+e \right)}^{th}}\] term, we get the following as the result
\[\begin{align}
  & ={}^{21}{{C}_{2r+e-1}}{{\left( 1 \right)}^{2r+e-1}}\times {{\left( x \right)}^{21-(2r+e-1)}} \\
 & ={}^{21}{{C}_{2r+e-1}}{{\left( x \right)}^{21-(2r+e-1)}} \\
\end{align}\]
Now, similarly, we can use the formula given in the hint to write the \[{{\left( 3+4 \right)}^{th}}~\] term that is nothing but the 7th term of the expansion which is as follows
\[\begin{align}
  & ={}^{21}C{{}_{7-1}}\times {{\left( 1 \right)}^{7-1}}\times {{\left( x \right)}^{21-(7-1)}} \\
 & ={}^{21}C{{}_{6}}{{\left( x \right)}^{15}} \\
\end{align}\]
Now, these two terms are equal to each other, hence, we can compare the coefficients as follows
\[{}^{21}{{C}_{2r+e-1}}={}^{21}{{C}_{6}}\]
Now, on using the property of the binomial coefficients as follows
\[{}^{n}{{C}_{r}}={}^{n}{{C}_{r}}\ or{}^{n}{{C}_{n-r}}\]
Hence, on using this, we get

\[\begin{align}
  & 2r+e-1=21-6 \\
 & 2r=16-e \\
 & r=\dfrac{16-e}{2} \\
\end{align}\]
Hence, the value of r is \[\dfrac{16-e}{2}\] .

NOTE: -
The students can make an error if they don’t know the formula or the property of binomial coefficients as follows
\[{}^{n}{{C}_{r}}={}^{n}{{C}_{r}}\ or{}^{n}{{C}_{n-r}}\]
(Which means two binomial coefficients can be equal if they are of the above two forms)
Without this property, solving the question is not possible.