Independency in Multinomial

Probability Level 4

( x 2 − 2 + 1 x 2 ) n \large \displaystyle \left( x^2 - 2 + \dfrac{1}{x^2} \right)^n

The total number of terms which are dependent on the value of x x in the expansion of above term.

n + 1 n + 1 2 n + 1 2n + 1 n 2 n^2 2 n 2 2n^2 2 n 2n n n None of these

Akhil Bansal by Akhil Bansal , 22, India

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1 solution

Akhil Bansal
Nov 11, 2015

⇒ ( x 2 − 2 + 1 x 2 ) n \Rightarrow \displaystyle \left( x^2 - 2 + \dfrac{1}{x^2} \right)^n ⇒ ( ( x − 1 x ) 2 ) n = ( x − 1 x ) 2 n \Rightarrow \left( \left( x - \dfrac{1}{x}\right)^2\right)^n =\color{#0C6AC7}{ \left(x - \dfrac{1}{x}\right)^{2n}}

General term of given expansion, T r + 1 = ( 2 n r ) ⋅ x 2 n − r ⋅ ( − 1 ) r x − r T_{r + 1} = \dbinom{2n}{r}\cdot x^{2n -r} \cdot (-1)^r x^{-r}
Total number of terms in given expansion is 2 n + 1 \color{#D61F06}{ 2n + 1} .

For term independent of x, 2 n − 2 r = 0 ⇒ n = r 2n - 2r = 0 \Rightarrow n = r

Therefore, only 1 \color{#3D99F6}{1} term is independent of x x

Hence, Number of terms dependent on x x = 2 n + 1 − 1 = 2 n \color{#D61F06}{2n + 1} - \color{#3D99F6}{1} = \boxed{ 2n}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

You should clarify the phrasing. Does the expression x + 2 x x + 2x have 2 terms which are independent of x x ? Or just 1?

Nice solution! Did it the same way :)

B.S.Bharath Sai Guhan - 5 years, 7 months ago

You should clarify the phrasing. Does the expression x + 2 x x + 2x have 2 terms which are independent of x x ? Or just 1?

Calvin Lin Staff - 5 years, 6 months ago

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