Q.1.7 - A First Course in Probability

Question

 (a) In how many ways can 3 boys and 3 girls sit in a row? (b) In how many ways can 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together? (c) In how many ways if only the boys must sit together? (d) In how many ways if no two people of the same sex are allowed to sit together? 

Step-by-Step Solution

Answer

Part(a)  No. of ways can 3 boys and 3 girls sit in a row   is  720

Part (b) No. of ways can 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together  is 72

Part (c)  No. of ways if only the boys must sit together  is 144

Part (d)  No. of ways if no two people of the same sex are allowed to sit together  is 72

Step 1 part (a). Given information

we have to find out number of ways can 3 boys and 3 girls sit in a row 

Step 2 Part (a). Finding the number of ways

Here 3 boys and 3 girls sit in a row, so there are 6 different people, hence the number of 6 different ways to sit is $6!=120$

Step1 . Part(b). Given information

We need to find out no. of ways  where 3 boys and 3 girls sit in a row if the boys and the girls are each to sit together 

Step 2 Part(b) .Finding the number of ways if boys and girls are each to sit together

  If boys and girls sit together,   there will be 2 groups and each group is arranged in $2!=2$ ways.

Among 3 boys, we can arrange them in 3!=6 ways, similarly we can arrange girls in $3! = 6$ways.

Hence the number of ways if  boys and girls are each to sit together are $2\times 6\times 6=72$

Step 1 .Part (c) .Given information

We have to find out ways if only the boys must sit together 

Step2 Part (C). Finding the number of ways if only the boys must sit together

If the boys are sitting together, then the number of boys and girls will be 4 different objects and we can arrange in $4!=24$ ways. The arrangement within-group is $3! =6$ ways.

Then the  number of ways if only the boys must sit together =$24\times 6=144$

Step 1 Part (d) .Given information

We have to find a number of ways if no two people of the same sex are allowed to sit together 

Step 2 Part (d) Finding ways if no two people of the same sex are allowed to sit together

Suppose that if a  girl is sitting first place, For the first place the possible number of ways from girls group can sit is 3 , after this girl sitting in third place, the number of different ways is 2  and the next sitting place for a girl is the fifth place and the number of ways is 1. Thus the Total number of ways girls can be in different places are $3\times 2\times 1=6$.

Similarly, we can arrange boys in second, fourth, and sixth places. then the number of ways boys can be in different places are $3\times 2\times 1=6$.

Thus ways if the number of girls and boys together is $6\times 6=36$ where the girl is sitting in the first place.

Similarly, if the boy is  sit in place, then the number of ways girls and sit together is $6\times 6=36$

Thus if no two people of the same sex are allowed to sit together  or sit alternatively is$36+36=72$