Question

Find number of arrangements of 4 letters taken from the word EXAMINATION.

• A. $2454$
• B. $2500$
• C. $2544$
• D. None of these

Answer 1

The correct option is A $2454$

In the word $EXAMINATION$ there are $8$ distinct words.

$Case1:$ When all words are distinct

We can take $4$ words from $8$ distinct in ${}^8{C}_4$ and can arrange the $4$ words in $4!$ ways.

$\Rightarrow$ Total number of ways $={}^8{C}_4\times 4!$

$=\frac{8\times 7\times 6\times 5}{4\times 3\times 2\times 1}\times 4\times 3\times 2\times 1$

$=1680$

$Case2:$ When $1$ word is alike and $2$ is distinct.

There are $3$ group of alike words. We can choose in ${}^3{C}_1$ ways and the distinct words in ${}^7{C}_2$.

We can arrange the word in $4!$ ways but two word are same so $\frac{4!}{2!}$ ways

$\Rightarrow$ Total number of ways $={}^7{C}_2\times {}^3{C}_1\times \frac{4!}{2!}$

$=756$

$Case3:$ Two are alike

Total number of ways $={}^3{C}_2\times \frac{4!}{2!}\times 2!$

$=18$

Total number of ways in which $4$ letter words can be made out of the word $EXAMINATION$

$\Rightarrow$ $1680+756+18$

$\Rightarrow$ $2454$