six girls are to enter a dance with ten boys so that every girl is between two boys. what is the probability that some specified boy remains between two boys?
Thank you. But sorry the question is #six girls are to enter a dance with ten boys to form a ring...#
The ten boys created 10 successive intervals (arcs). 6 girls can choose six of the intervals in C(10, 6) ways.
If a particular boy is to be between 2 boys, the number of intervals (arcs) reduces by 2 to 8 and the choice now is C(8, 6) ways
So, probability =C(8, 6)/C(10, 6)=2/15
To find the probability that a specified boy remains between two boys, we need to determine the total number of possible arrangements where this condition is satisfied, and then divide it by the total number of possible arrangements.
First, let's calculate the total number of possible arrangements for the boys and girls. Since there are 10 boys and 6 girls, we have a total of (10 + 6) = 16 individuals in the dance.
To satisfy the condition that every girl is between two boys, we can think of it as positioning the girls first and then filling the spaces between them with boys. Let's consider arranging the girls in a row first:
_ G _ G _ G _ G _ G _
There are 6 girls, and we need to choose 6 out of the 10 spaces between them. This can be done in 10C6 ways, which is calculated as:
10C6 = 10! / (6! * (10 - 6)!) = 210
So we have 210 possible ways to arrange the girls.
Now, we need to fill the remaining spaces between the girls with the 10 boys. Since there are 10 boys and 5 spaces between the 6 girls (the outermost spaces do not need to be filled as they are not between two girls), we have the following distribution of boys:
_ G _ G _ G _ G _ G _
Now, we need to calculate the number of ways to arrange the boys in these spaces. This can be calculated using the multinomial coefficient, as there are multiple spaces to be filled.
The multinomial coefficient is given by:
(n!) / (k1! * k2! * ... * km!)
Where n is the total number of objects to be arranged, and k1, k2, ..., km represent the number of identical objects of different types.
In our case, we have 10 boys to be arranged between the girls. So applying the multinomial coefficient formula, we get:
10! / (1! * 1! * 1! * 1! * 1! * 5!)
= 10! / (5!)
= 10 * 9 * 8 * 7 * 6
= 30,240
Therefore, there are 30,240 possible ways to arrange the boys between the girls.
Now, to find the probability that a specified boy remains between two boys, we need to consider those arrangements where that condition is satisfied. Since we have 2 spaces between each pair of girls, there are 5 pairs in total. One boy will be chosen to remain between two boys out of the 10 available boys.
Therefore, the probability is:
1 / 30,240 ≈ 0.0000330767731
So, the probability that some specified boy remains between two boys is approximately 0.000033 or 0.0033%.
First let arrange the 10 boys.
This can be done in 10! ways
Now there are 9 places in this arrangement where we can place a girl, but we only have 6 girls.
So we have to choose 6 of those 9 places to insert the single girls.
There are C(9,6) or 84 ways of doing so
But we can also arrange the girls in 6! ways to fit into those 84 places,
So I get 84(10!)(6!) ways of doing this
= rather large , I get 2.1947 x 10^11