Question

Find the sixth term of expansion of
(y^1/2 + x^1/2) , if the binomial
cefficient of the third term from
the end is 45.
​

Answer 1

➜Given :

•The binomial coefficient of the third term from the end =45

• Given expansion =$\bf{(y^{\dfrac{1}{2}}+x^{\dfrac{1}{2}})}$

➜To Find :

•The sixth term of expansion of $\bf{y^{\dfrac{1}{2}}+x^{\dfrac{1}{2}}=?}$

➜Solution :

⇒Given expressions is $\red{\bf{y^{\dfrac{1}{2}}+x^{\dfrac{1}{2}}}}$

⇒Given that ,

•Binomial coefficient of the third term from the end =45

$\\ \implies\sf{^{n}C_{n-2}=45}$

$\\ \implies\sf{^{n}C_{2}=45}$

$\\ \implies\sf{\dfrac{n(n-1)(n-2)!}{2!(n-2)!}=45}$

$\\ \implies\sf{n(n-1)=90}$

$\\ \implies\sf{n^{2}-n-90=0}$

$\\ \implies\sf{(n-10)(n+9)=0}$

$\\ \implies\sf{n=10 \: \: \quad \: n=-9}$

Now, sixth term

$\\ \implies\sf{^{10}C_{5}(y^{\dfrac{1}{2}})^{10-5}(x^{\dfrac{1}{3}})^{5}}$

$\\ \implies\sf{225y^{\dfrac{5}{2}}.x^{\dfrac{5}{3}}}$

•the sixth term is

$\\ \red\bigstar\boxed{\sf{225y^{\dfrac{5}{2}} . x^{\dfrac{5}{3}}}}$