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.

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