JEE Main 2021 (Online) 3rd August Evening ShiftQuestion 1
In the Binomial expansion of $\left(2x\sqrt{x}+3y^{\frac{1}{16}}\right)^{33}$, the number of terms, having positive integral powers of $x$ and $y$ is :
Binomial Theorem · Rational and Integral Terms
Solution
Concept. Write both powers in terms of $r$ and check which $r$ make both positive integers. Formulas used. (1) $T_{r+1}={}^{n}C_r\,a^{n-r}b^r$, with $2x\sqrt x=2x^{3/2}$Step 1 - Powers. In $T_{r+1}$ the power of $x$ is $\frac{3(33-r)}{2}$ and of $y$ is $\frac{r}{16}$. Step 2 - Power of $y$. A positive integer with $0\le r\le33$ needs $r=16$ or $r=32$. Step 3 - Power of $x$. $r=16$: $\frac{3\cdot17}{2}=\frac{51}{2}$. $r=32$: $\frac32$. Neither is an integer. Step 4 - Count. No such term: $0$. Answer: (A)