Question 16

Consider all $$4$$ element subsets of the set $$A=\{1,2,3,\ldots,8\}$$. Each of these subsets has a greatest element. The arithmetic mean of the greatest elements of these $$4$$ element subsets is

Correct Answer: 7.2

Solution

There are $$\binom84=70$$ subsets. The numbers of subsets whose greatest elements are $$8,7,6,5,4$$ are respectively $$\binom73,\binom63,\binom53,\binom43,\binom33$$. Hence the mean is $$\frac{8\binom73+7\binom63+6\binom53+5\binom43+4\binom33}{70}=\frac{504}{70}=7.2$$.

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Consider all $$4$$ element subsets of the set $$A=\{1,2,3,\ldots,8\}$$. Each of these subsets has a greatest element. The arithmetic mean of the greatest elements of these $$4$$ element subsets is

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