# Math Problem

#### dennisDDS

10+ Year Member

Five chairs are lined up in a row. How many ways can three boys and two girls sit in these chairs if the girls always sit side by side?

The final answer is 48.. but is there a better way to streamline the calculation.

#### Pwn

10+ Year Member
Ya think about it like this:

5 seats and 2 girls have to be together how many ways can the group of two girls be arranged (not counting the fact that the two girls are different)
this gives 4 possibilities
xx000 0xx00 00xx0 000xx (where x's are girls 0's are guys)

But there are two different girls so we have twice as many ways to arrange the girls!
so 8 possibilities

now the boys are all different, how many ways can we arrange the three boys (6 ways or 3!)

thus 6*8 = 48 possibilities

if this doesnt make sense its prolly cuz im high

#### dennisDDS

10+ Year Member
ahh ok.. so the girls show 4 different arrangements but since each girl can switch it results in 4x2 = 8

for the men they are 2 different arrangements but since each of the 3 can switch it results in 3x2 = 6

thus, the total arrangement = 8 x 6 or 48..

did i get that right?

#### Pwn

10+ Year Member
ahh ok.. so the girls show 4 different arrangements but since each girl can switch it results in 4x2 = 8
I only get this part of what you're saying (other part might be right but I dont get it). Basically what I'm saying is that you get those 4 arrangements (demonstrated by the x's and o's) and then accounting for the fact that your x's are 2 different girls you get two possibilities for each of the 4 arrangements (resulting in 8). Now you do the same thing for the boys (you have 3 o's but 3 diff boys!) and using the same logic you get 6 different ways for each of the 8 arrangements. this results in 6*8 =48 ways.

hope that helps

10+ Year Member
4!*2! = 48

#### dennisDDS

10+ Year Member
thanks it helps. #### jefff

5+ Year Member
best way to think about this is how many ways can they be arranged if the girls and guys could sit in any of the seats (5!=5x4x3x2x1=120)

so the next question (the real original questions)

How many ways (out of all the possible arrangements) can three boys and two girls sit in these chairs if the girls always sit side by side?

you have 5 people lets call them units...

2 g's(g & g) and 3 b's (g g b b b) **bolded a g to keep track of it**

since the g's need to be next to each other it will be better to think of it like this like:

gg b b b (as in 4 units instead of 5)

the number of ways the g units can be arranged is 4! (4x3x2x1=24) but since the g units can swap (gg -> gg) the answer is now

2x4! = 48