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Find the Ratio of the Coefficients of Xp and Xq in the Expansion of ( 1 + X ) P + Q . - Mathematics

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Question

Find the ratio of the coefficients of xp and xq in the expansion of \[\left( 1 + x \right)^{p + q}\] .

 

Solution

Coefficient of xp in the expansion of  \[\left( 1 + x \right)^{p + q}\] is  \[{}^{p + q} C_p\] .

Coefficient of xq  in the expansion of \[\left( 1 + x \right)^{p + q}\]  is \[{}^{p + q} C_q\]
Now,
\[\frac{{}^{p + q} C_p}{{}^{p + q} C_q} = \frac{\frac{\left( p + q \right)!}{p!q!}}{\frac{\left( p + q \right)!}{q!p!}} = 1\]
Hence, the ratio of the coefficients of xp and xq in the expansion of \[\left( 1 + x \right)^{p + q}\]  is 1 : 1.
 
 
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Proof of Binomial Therom by Combination
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Chapter 18: Binomial Theorem - Exercise 18.3 [Page 45]

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RD Sharma Mathematics [English] Class 11
Chapter 18 Binomial Theorem
Exercise 18.3 | Q 13 | Page 45
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