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If the number of terms in \[{{\left( 1+{{x}^{-1}}+{{x}^{-2}} \right)}^{n}}\] is 53, then the largest prime p so that \[n!\] is divisible by \[{{5}^{p-1}}\] is
(a) 3
(b) 5
(c) 7
(d) 11

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Last updated date: 20th Jun 2024
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Answer
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Hint: We solve this problem by using the standard result that is the number of terms in the expansion of \[{{\left( A+\dfrac{B}{x}+\dfrac{C}{{{x}^{2}}} \right)}^{n}}\] is given as \[2n+1\] then we find the value of \['n'\]
After that we find the power of 5 that includes in \[n!\] which means the largest power of 5 such that \[n!\] is divisible by \[{{5}^{p-1}}\] to get the value of ‘p’.

Complete step by step answer:
We are given that the expansion of \[{{\left( 1+{{x}^{-1}}+{{x}^{-2}} \right)}^{n}}\] has 53 terms.
We know that the number of terms in the expansion of \[{{\left( A+\dfrac{B}{x}+\dfrac{C}{{{x}^{2}}} \right)}^{n}}\] is given as \[2n+1\]
By using the above result to given condition we get
\[\begin{align}
  & \Rightarrow 2n+1=53 \\
 & \Rightarrow n=26 \\
\end{align}\]
Now, let us find the multiples of 5 that are involved in \[26!\]
We know that
\[n!=1\times 2\times 3\times 4..........\times n\]
By using the above formula we get
\[\Rightarrow 26!=1\times 2\times 3\times .........\times 26\]
Now, let us write the 5 multiples separately then we get
\[\Rightarrow 26!=\left( 5\times 10\times 15\times 20\times 25 \right)\times k\]
Here, we can assume that the remaining product as \['k'\]
Now let us write the above equation in the power of 5 then we get
\[\Rightarrow 26!={{5}^{6}}\left( p \right)\]
Here, we can assume that the remaining product as \['p'\]
Now, we can see that the highest power of 5 that divides \[26!\] exactly is \[{{5}^{6}}\]
We are given that \[n!\] is divisible by \[{{5}^{p-1}}\] where \[n=26\]
Now, by comparing the given statement with the result we get
\[\begin{align}
  & \Rightarrow {{5}^{p-1}}={{5}^{6}} \\
 & \Rightarrow p-1=6 \\
 & \Rightarrow p=7 \\
\end{align}\]
Therefore the largest prime number ‘p’ such that \[n!\] is divisible by \[{{5}^{p-1}}\] is 7

So, the correct answer is “Option c”.

Note: Students may make mistakes in taking the formula of number of terms of expansion.
The number of terms in the expansion of \[{{\left( A+\dfrac{B}{x}+\dfrac{C}{{{x}^{2}}} \right)}^{n}}\] is given as \[2n+1\]
We also have other formula that is
The number of terms in the expansion of \[{{\left( A+B+C \right)}^{n}}\] is given as \[\left( n+1 \right)\left( n+2 \right)\]
Students may get confused between these two formulas and get the wrong answer.