What does it 'Mean'?

Probability Level 2

We have a set of interest, S S ={ 1 , 2 , 3 , . . . . , 1000 1,2,3,....,1000 }. Find the arithmetic mean of the smallest elements, F ( 1000 , 90 ) F(1000,90) of subsets of S of cardinality 90.

For example, if we take a set A A ={ 1 , 2 , 3 , 4 1,2,3,4 }, the subsets of A A of cardinality 2 are { 1 , 2 1,2 }; { 1 , 3 1,3 };{ 1 , 4 1,4 }; { 2 , 3 2,3 }; { 2 , 4 2,4 }; { 3 , 4 3,4 }. F ( 4 , 2 ) F(4,2) = 3 × 1 + 2 × 2 + 1 × 3 1 + 2 + 3 \frac{3×1+2×2+1×3}{1+2+3} = 5 3 \frac{5}{3}


The answer is 11.

Alapan Das by Alapan Das , 20, India

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1 solution

Alapan Das
May 16, 2019

F ( n , r ) = A / B F(n,r)=A/B A = 1 × [ ( n − 1 ) C ( r − 1 ) ] + 2 × [ ( n − 2 ) C ( r − 1 ) ] + . . . ( n − r + 1 ) × [ ( r − 1 ) C ( r − 1 ) = ( n + 1 ) / ( r + 1 ) ( n C r ) A={1×[(n-1)C(r-1)]+2×[(n-2)C(r-1)]+...(n-r+1)×[(r-1)C(r-1)}={(n+1)/(r+1)}(nCr)

B = [ ( n − 1 ) C ( r − 1 ) ] + [ ( n − 2 ) C ( r − 1 ) + . . . . . . . . . . . . + [ ( r − 1 ) C ( r − 1 ) ] = n C r B=[(n-1)C(r-1)]+[(n-2)C(r-1)+............+[(r-1)C(r-1)]=nCr

A / B = n + 1 r + 1 A/B=\frac{n+1}{r+1} . In the given question n = 1000 , r = 90 n=1000, r=90 . F ( 1000 , 90 ) = 1001 / 91 = 11 F(1000,90)=1001/91=11 .

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