Question

Ten IIT and $2DCE$ students sit in a row. The number of ways in which exactly $3$ IIT students sit between $2DCE$ students is

• A. ${}^{10}{C}_3\times 2!\times 3!\times 8!$
• B. $10!\times 2!\times 3!\times 8!$
• C. $5!\times 2!\times 9!\times 8!$
• D. None of these

Answer 1

The correct option is A ${}^{10}{C}_3\times 2!\times 3!\times 8!$

Three IIT students who will be between the IIT students can be selected in ${}^{10}{C}_3$ ways.
Now, two DCE students having three IIT students between them can be arranged in $2!\times 3!$ ways.
Finally, a group of above five students and the remaining seven students together can be arranged in $8!$ ways.
Hence, total number of ways is ${}^{10}{C}_3\times 2!\times 3!\times 8!$