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Binomial Theorem

Section: Mathematics

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Question : 9 of 33

Marks: +1, -0

The terms containing

x^r y^s

r

s

) are present in both the expansions of

(x+y^2)^{13}

(x^2+y)^{14}

α

is the number of such terms, then the sum

α \sum\limits_{r, s}(r+s)=

[AP EAPCET 22 May 2025 S2]

$27$
$40$
$18$
35

Solution:

General term for

(x+y^2)^{13}

={}^{13}C_n x^{13-n} y^{2n}

The general term for

(x^2+y)^{14}

={}^{14}C_{r_2} x^{2r_2} y^{14-r_2}

Equate the power of

x

and

y

\;13-r_1=2 r_2

\;2 r_1=14-r_2

Solving these equation, we get

r_1=5, r_2=4

The common term is

x^{13-5} y^{2(9)}=x^8 y^{10}

The common term is

x^8 y^{10}

\Rightarrow \;\; r=8, s=10

α=1

, since there is only one common term

\;\Rightarrow \;\; α(r+s)=1(8+10)=18

\;\therefore \;\text{ Sum }\; α \sum\limits_{r,s}(r+s)=18

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