Question

In how many ways can $17$ persons depart from railway station in $2$ cars and $3$ autos, given that $2$ particular persons depart by same car ($4$ persons can sit in a car and $3$ persons can sit in an auto)?

• A. $\frac{15!}{2!4!(3!{)}^3}$
• B. $\frac{16!}{(2!{)}^{2}4!(3!{)}^3}$
• C. $\frac{17!}{2!4!(3!{)}^3}$
• D. $\frac{15!}{4!(3!{)}^3}$

Answer 1

The correct option is D $\frac{15!}{4!(3!{)}^3}$

Make $1$ group of $2$ persons, $1$ group of $4$ persons and $3$ groups of $3$ persons among $15$ persons (except $2$ particular persons). Hence the number of ways by grouping method is
$\frac{15!}{2!4!(3!{)}^{3}3!}$
Now we add that $2$ persons in the group of $2$ persons and thus number of arrangements of these groups into cars and autos is
$\frac{15!}{2!4!(3!{)}^{3}3!}\times 2!\times 3!=\frac{15!}{4!(3!{)}^3}$