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Algebra Level 3

What is the co-efficient of x 53 { x }^{ 53 } in the expansion ∑ m = 0 100 ( 100 m ) ( x − 3 ) 100 − m . 2 m \displaystyle \sum _{ m=0 }^{ 100 }{ \begin{pmatrix} 100 \\ m \end{pmatrix} } { (x-3) }^{ 100-m }{ .2 }^{ m } ?

Here ( a b ) \begin{pmatrix} a \\ b \end{pmatrix} means the number of ways of choosing b b objects from a a .

( 100 53 ) \begin{pmatrix} 100 \\ 53 \end{pmatrix} 0 0 − ( 100 53 ) -\begin{pmatrix} 100 \\ 53 \end{pmatrix} None of the above.

Chirag Trasikar by Chirag Trasikar , 24, India

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1 solution

Prasun Biswas
Feb 12, 2015

Using binomial theorem, we can see that the sum is simply ( x − 3 + 2 ) 100 (x-3+2)^{100} ,i.e., ( x − 1 ) 100 = ( 1 − x ) 100 (x-1)^{100}=(1-x)^{100}

Now, we note that the term with x 53 x^{53} is the 5 4 th 54^{\textrm{th}} term in the expansion of the given expression. And using the formula for the term of a binomial, we have,

T r + 1 = ( 100 r ) ( 1 ) 100 − r ( − x ) r ⟹ T 54 = − ( 100 53 ) x 53 T_{r+1}=\binom{100}{r}(1)^{100-r}(-x)^{r}\implies T_{54}=-\binom{100}{53}x^{53}

where, T x T_x denotes the x t h x^{th} term of the expansion.

Hence, Required coefficient = − ( 100 53 ) =-\dbinom{100}{53}

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