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Probability Level 3

Define a set X = { 1 , 2 , 3 , 4 , 5 , 6 } X=\{1,2,3,4,5,6\} . Also define a function f ( x ) = ( x + a ) ( x + b ) ( x + c ) f(x)=(x+a)(x+b)(x+c) , where a a , b b , and c c are distinct values chosen from the set X X .

Find the number of possible distinct values for the coefficient of x 2 x^2 in f ( x ) f(x) .

You can try more of my fundamental problems here .


The answer is 10.

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Donglin Loo by donglin loo , 22, Singapore

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1 solution

Donglin Loo
Jun 19, 2018

Upon expansion,

the coefficient of x 2 = a + b + c x^2=a+b+c

Minimum value of a + b + c = 1 + 2 + 3 = 6 a+b+c=1+2+3=6

Maximum value of a + b + c = 4 + 5 + 6 = 15 a+b+c=4+5+6=15

∴ \therefore the number of possible distinct values for the coefficient of x 2 = 15 − 6 + 1 = 10 x^2=15-6+1=10

It is not hard to see that the values between 6 6 to 15 15 are all attainable. We can alter the values starting from the minimum by progressively increasing 1 1 selectively.

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