In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 15 boys and 7 girls are competing, how many different ways could the six medals possibly be given out?
To determine the number of different ways the six medals can be distributed, we can think of it as a combination problem.
First, let's consider the boys. There are 15 boys competing, and we need to determine the number of ways to select 3 out of the 15 for the medals. We can use the concept of combinations to solve this.
The number of ways to select r objects out of a set of n objects is given by the formula: C(n, r) = n! / (r!(n-r)!)
In this case, n = 15 (number of boys) and r = 3 (number of medals). So, we have C(15, 3).
C(15, 3) = 15! / (3!(15-3)!) = 15! / (3!12!) = (15 * 14 * 13) / (3 * 2 * 1) = 455
Thus, there are 455 different ways to award the medals to the boys.
Next, let's consider the girls. There are 7 girls competing, and we need to determine the number of ways to select 3 out of the 7 for the medals. Again, we can use the combination formula.
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
So, there are 35 different ways to award the medals to the girls.
Now, to determine the total number of different ways to distribute all six medals, we multiply the number of ways for boys and girls together.
Total number of ways = 455 * 35 = 15,925.
Therefore, there are 15,925 different ways that the six medals can possibly be given out.