JEE Main 2025 (Online) 23rd January Morning ShiftQuestion 1
The sum of all rational terms in the expansion of $\left(1+2^{1/3}+3^{1/2}\right)^{6}$ is equal to $\_\_\_\_\_\_\_\_$ .
Binomial Theorem · Multinomial Expansion · Rational and Integral Terms
Answer$612$
Solution
Concept. Use the multinomial expansion; a term is rational when the power of $2^{1/3}$ is a multiple of $3$ and the power of $3^{1/2}$ is even. Formulas used. (1) $(p+q+s)^n=\sum\dfrac{n!}{a!\,b!\,c!}\,p^aq^bs^c$, with $a+b+c=n$Step 1 - General term. $$\frac{6!}{a!\,b!\,c!}\,2^{b/3}\,3^{c/2},\qquad a+b+c=6.$$ It is rational iff $3\mid b$ and $2\mid c$. Step 2 - List the terms. $b=0$: $c=0$ gives $1$; $c=2$ gives $15\cdot3=45$; $c=4$ gives $15\cdot9=135$; $c=6$ gives $27$. $b=3$: $c=0$ gives $20\cdot2=40$; $c=2$ gives $60\cdot2\cdot3=360$. $b=6$: $c=0$ gives $2^2=4$. Step 3 - Add. $1+45+135+27+40+360+4=612$. Answer: 612