# What's the correct answer: 7 boys 7 girls with clones

Q: 7 boys, 3 are identical clones. 7 girls, 2 are clones. one of the non-clone boy is Kevin, how many ways can Kevin sit in a row so he is not next to a non-clone girl ?

My A: total - (Kevin sits with a non-clone girl). = total - (Kevin has non-clone girl in the left + Kevin has non-clone girl in the right - both sides of Kevin are non-clone girls). (updated based on comment)

But my answer is different from others. What's the correct answer pls ?

The question asks for ways they can sit such that Kevin is "not next to a non-clone girl." The phraseology is a little bit convoluted, but it sounds like you have to subtract the cases Kevin is next to a non-clone girl. You subtracted the cases Kevin is next to a clone girl.

Other than that, your method is sound.

• you are right, I miss typed, thx ! – user3552178 Jan 7 '18 at 21:19

It is easier to solve directly !

Kevin, the $3$ other distinct boys, the $3$ boy clones and the $2$ girl clones can be seated in $\dfrac{9!}{3!2!}$ ways.

Now erase their identity except for Kevin, e.g. $\circ\circ\circ\; K\circ\circ\circ\circ\circ\;$ or, say, $K\circ\circ\circ\circ\circ\circ\circ\circ\;$

Wherever Kevin is there in such a row, there are two spots on either side of $K$ where distinct girls can't be placed, so they can be successively placed in the row in $8\cdot9\cdot10\cdot11\cdot12 = 95040$ ways,

giving an answer of $\left[\dfrac{9!}{3!2!}\times 95040\right]$

PS Have removed the confusion by terming non-clone girls as distinct girls !

• same answer, sounds great ! – user3552178 Jan 8 '18 at 13:28