JEE Main 2025 Jan 23 Shift 1, Mathematics Q22: Introduction to Binomial Expansion

By the multinomial theorem, a generic term is $\frac{6!}{r_1!r_2!r_3!}\cdot 1^{r_1}\cdot 2^{r_2/3}\cdot 3^{r_3/2}$ with $r_1+r_2+r_3=6$.

This term is rational only when $r_2$ is a multiple of 3 (so $r_2\in \{0,3,6\}$) and $r_3$ is even ($r_3\in \{0,2,4,6\}$).

Enumerate valid $(r_1,r_2,r_3)$ triples and their contributions:

$(6,0,0):\frac{6!}{6!}=1$

$(4,0,2):\frac{6!}{4!2!}\cdot 3=15\cdot 3=45$

$(2,0,4):\frac{6!}{2!4!}\cdot 9=15\cdot 9=135$

$(0,0,6):\frac{6!}{6!}\cdot 27=27$

$(3,3,0):\frac{6!}{3!3!}\cdot 2=20\cdot 2=40$

$(1,3,2):\frac{6!}{1!3!2!}\cdot 2\cdot 3=60\cdot 6=360$

$(0,6,0):\frac{6!}{6!}\cdot 4=4$

Sum $=1+45+135+27+40+360+4=612$.