From 10 numbers a,b,c,...j all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the 10 numbers?

posader86

posader86

Answered question

2022-07-23

From 10 numbers a , b , c , . . . j all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the 10 numbers?

Answer & Explanation

encoplemt5

encoplemt5

Beginner2022-07-24Added 15 answers

There are C ( 10 , 4 ) = 10 ! 4 ! 6 ! different sets of 4 numbers chosen among x 1 , x 2 , x 10 . Each number x i belongs to C ( 9 , 3 ) = 9 ! 3 ! 6 ! such sets, because the other three numbers in the same set can be chosen in C ( 9 , 3 ) different ways. Hence the average on all sets is:
1 C ( 10 , 4 ) 1 i < j < k < l 10 x i + x j + x k + x l 4 = 1 4 C ( 10 , 4 ) i = 1 10 C ( 9 , 3 ) x i = C ( 9 , 3 ) 4 C ( 10 , 4 ) i = 1 10 x i = 1 10 i = 1 10 x i
and both averages are the same.
This also works in general for the case of all sets of n numbers chosen among N. The key is all numbers x i appear the same number of times in the final sum.

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