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Algebra Level 2

Let S = ( x − 1 ) 4 + 4 ( x − 1 ) 3 + 6 ( x − 1 ) 2 + 4 ( x − 1 ) + 1 \displaystyle S={ \left( x-1 \right) }^{ 4 }+{ 4\left( x-1 \right) }^{ 3 }{ +6\left( x-1 \right) }^{ 2 }+4\left( x-1 \right) +1 . Then S equals:

( x − 1 ) 4 \displaystyle {\left( x-1 \right) }^{ 4 } ( x + 1 ) 4 \displaystyle {\left( x+1 \right) }^{ 4 } x 4 \displaystyle { x }^{ 4 } none of these

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Soumo Mukherjee by Soumo Mukherjee , 25, India

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1 solution

Soumo Mukherjee
Dec 19, 2014

Consider the binomial expansion ( a + 1 ) 4 = a 4 + 4 a 3 + 6 a 2 + 4 a + 1 \displaystyle { \left( a+1 \right) }^{ 4 }={ a }^{ 4 }+4{ a }^{ 3 }+6{ a }^{ 2 }+4a+1 .Let ( x − 1 ) = a . \displaystyle \left( x-1 \right) =a. Then S = ( x − 1 + 1 ) 4 = x 4 \displaystyle S={ \left( x-1+1 \right) }^{ 4 }={ x }^{ 4 }

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Observing the binomial coefficients(1,4,6,4,1) gave me the idea. Nice problem!!

Hariharan Gandhi - 3 years, 11 months ago

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