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\[A. = \dfrac{{9!}}{{2!2!}} - \dfrac{{7!}}{{2!2!}}\]

\[B. = \dfrac{{9!}}{{3!2!}} - \dfrac{{7!}}{{2!2!}}\]

\[C. = \dfrac{{9!}}{{3!2!2!}} - \dfrac{{7!}}{{2!2!2!}}\]

\[D. = \dfrac{{9!}}{{3!2!2!}} - \dfrac{{7!}}{{2!2!}}\]\[\]

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Writing down the letters of \[ENGINEERING\]-

$

E,E,E \\

N,N,N \\

G,G \\

I,I \\

R \\

$

$E$ comes $3$ times, $N$ comes $3$ times, $G$ comes $2$ times, $I$ comes $2$ times & $R$ comes $1$ time.

Now, when $N's$ come together.

Take all $N$ as one, then total numbers of letters become $9$ and they arrange $9!$ ways.

But some repeated letters are present.

So, no. of ways arrangement needs when $N$ remarks together $ = \dfrac{{9!}}{{3!2!2!}}$.

When \[N's\] & \[E's\] are come together \[ = \dfrac{{7!}}{{2! \times 2!}}\].

\[\therefore \] Required number of arrangements are \[ = \left( {\dfrac{{9!}}{{3!2!2!}} - \dfrac{{7!}}{{2! \times 2!}}} \right)\].