Question

Eight persons decided to go for a picnic into two groups, one group to go by car and the other group by train. In how many ways can this be done if there must be atleast $3$ persons in each group ?

Answer 1

We have,

Total persons$=8$

No. of ways picnic $=2$

Group $=2$

At least $3$ persons in each group.

So, the groups

($3$ in car and $5$ in train)

($4$ in car and $4$ in train)

($5$ in car and $3$ in train)

So,

The number of ways they can go for a picnic,

${=}^8{C}_3{\times }^5{C}_5{+}^8{C}_4{\times }^4{C}_4{+}^8{C}_5{\times }^3{C}_3$

$\Rightarrow 56\times 1+70\times 1+56\times 1$

$\Rightarrow 182$

There are number of ways is $182$.

Hence, this is the answer.