I am missing "x"

Algebra Level 4

Find the coefficient of the term which does not contain " x x " in the expansion of ( x 1 3 − 1 x ) 15 . \large (x^{\frac{1}{3}}-\frac{1}{\sqrt{x}})^{15}.


The answer is 5005.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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3 solutions

Chew-Seong Cheong
Oct 31, 2014

( x 1 3 − 1 x ) 15 = ( x 1 3 − x − 1 2 ) 15 = x 1 3 × 15 ( 1 − x − 1 2 − 1 3 ) 15 (x^{\frac{1}{3}}-\dfrac {1}{\sqrt{x}})^{15} = (x^{\frac{1}{3}}-x^{-\frac {1}{2}} )^{15} = x^{\frac {1}{3} \times 15}(1-x^{-\frac {1}{2} -\frac {1}{3}} )^{15}

= x 5 ( 1 − x − 5 6 ) 15 = x 5 ∑ n = 0 15 ( 15 n ) x − 5 6 n = x^5(1-x^{-\frac {5}{6} })^{15} = x^5 \sum _{ n=0 }^{ 15 }{ \left( \begin{matrix} 15 \\ n \end{matrix} \right) { x }^{ -\frac { 5 }{ 6 } n } }

The term without x x is when n = 6 n = 6 and its coefficient = ( 15 6 ) = 15 × 14 × 13 × 12 × 11 × 10 1 × 2 × 3 × 4 × 5 × 6 = 5005 = \left( \begin {matrix} 15 \\ 6 \end {matrix} \right) = \dfrac {15\times 14\times 13\times 12 \times 11\times 10}{1\times 2\times 3\times 4\times 5\times 6} = \boxed{5005}

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I just used the formula T r + 1 = ( n r ) ( a ) n − r ⋅ ( b ) r T_{r+1}=\binom{n}{r} (a)^{n-r}\cdot (b)^r for the expansion ( a + b ) n (a+b)^n to find the term where x 0 x^0 is and then found the value of that term.

Prasun Biswas - 6 years, 6 months ago
Aman Sharma
Nov 4, 2014

By the binomial theoram genral term of the given expression is:- ( − 1 ) r ( 15 r ) x 5 − r 3 − r 2 (-1)^r \binom{15}{r}x^{5-\frac{r}{3}-\frac{r}{2}} Now for a term to not contain x,power of x in genral term must be zero ie 5 − r 3 − r 2 = 0 5-\frac{r}{3}-\frac{r}{2}=0 Solving it gives r=6 Putting r=6 in genral term gives 5005

4 Helpful 0 Interesting 0 Brilliant 0 Confused

At first, I thought that the cardinality of the term was asked, so I answered 7 7 , then I thought the value of r r was asked, so I answered 6 6 . Finally I realized the value of the term itself was asked, so I got the correct answer on the last try.

Prasun Biswas - 6 years, 6 months ago

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Same here. The question needs to be checked for phrasing.

Shashank Rammoorthy - 5 years, 11 months ago

9 / 3 − 6 / 2 = 0. S o i t i s 15 − 9 = 6. S o t h e c o e f f i c i e n t i s 15 ! 6 ! ∗ 9 ! = 5005. 9/3-6/2=0.~~So ~it~is~15-9=6. \\ So ~the~coefficient~is~~\dfrac{15!}{6!*9!}=5005.

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