Question

The new ointment was applied to four locations, and a control was applied to the other four. How many different choices were there for the four locations to apply the new ointment?

Answer 1

there are 70 possible choices for the four locations to apply the new ointment

Explanation:

Since we have a total of 8 locations ( 4 to the new ointment and 4 to the control) , each one can be chosen and since the order of the locations that are chosen for the new ointment is not relevant , then we know that the number of choices is given by the number of combinations of 4 elements in 8

number of combinations = 8 possible locations to the first ointment * 7 possible locations to the second ( since the first one was already located) * 6 to the third * 5 locations for the fourth / number of times the same combination is repeated ( the same locations but in different positions) = 8*7*6*5 / (4 possible positions for the first ointment* 3 possible positions to the second ointment (since the first one was already located * 2 possible positions of the third * 1 possible position of the fourth)

therefore

number of combinations = 8*7*6*5/(4*3*2*1 ) = 8!/((8-4)!*4!) = 70 possible combinations

thus there are 70 possible choices for the four locations to apply the new ointment