Skywalker wrote:In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?
The [spoiler]OA = 72[/spoiler] but I chose 24 since I added did the following:-
{ 3x2x1 ( boys ) + 3x2x1 ( girls ) } x 2 = 24.
while it should have been multiplication instead of "+".
When do we ADD & when do we multiply ? Please let me know.
When we're combining from different sources -- in this case, from our source of boys and from our source of girls -- we multiply the number of choices we have from each source:
Number of ways to arrange the 3 boys in the first 3 seats: 3*2*1 = 6
Number of ways to arrange the 3 girls in the last 3 seats: 3*2*1 = 6
From our source of boys we have 6 choices, from our source of girls we have 6 choices. To combine the number of choices we have from each source, we multiply:
6*6 = 36.
We also could reverse the order so that the girls are in the first 3 seats and the boys are in the last 3 seats:
Number of ways to arrange the 3 girls in the first 3 seats: 3*2*1 = 6
Number of ways to arrange the 3 boys in the last 3 seats: 3*2*1 = 6
From our source of girls we have 6 choices, from our source of boys we have 6 choices. To combine the number of choices we have from each source, we multiply:
6*6 = 36.
Now we need to count -- not combine -- all the possible ways to arrange the students. The first scenario gave us 36 ways, the second gave us another 36 ways, so we add: 36+36 = 72.
(Please note that if we realized that reversing the boys and the girls would give us another 36 ways, we could simply say that there are 2*36=72 total ways. This multiplication, however, is just a quicker way of adding the 36 ways from the first scenario to the 36 ways from the second scenario.)
Does this help?
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