Question

If $2$ are alike and the other $2$ distinct

• A. $1764$
• B. $2511$
• C. $2500$
• D. $3087$

Answer 1

The correct option is A $1764$

We need to form words of $4$ letters such that $2$ letters of same kind $\xi$ $2$ different.

We have $4$ letters that repeat $(I,S,T,L).$

$\Rightarrow$ We must choose only one of them $={}^4{C}_1=4$

$\Rightarrow$ Now excluding the chosen letter we have $7$ distinct letters out of which we must choose $2={}^7{C}_2=21$

$\Rightarrow$ Total $={}^4{C}_1\times {}^7{C}_2=4\times 21=84.$

$\Rightarrow$ Now these chosen letters can be rearranged in $\frac{4!}{2!}$ ways $=\frac{24}{2}=12$

$\Rightarrow$ Required $=84\times 21=1764.$

Hence, the answer is $1764.$