The sum of all rational terms in the expansion of $\left(1+2^{1 / 3}+3^{1 / 2}\right)^{6}$ is equal to

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Binomial Theorem

The sum of all rational terms in the expansion of

(1 + 2^{1/3} + 3^{1/2})^{6}

is equal to

Answer: 612

Using the multinomial expansion,

(1 + 2^{1/3} + 3^{1/2})^{6} = \sum \frac{6 !}{r _{1} ! r _{2} ! r _{3} !} 2^{r_{2} /3} 3^{r_{3} /2}, r_{1} + r_{2} + r_{3} = 6

A term is rational when

r_{2}

is a multiple of

3

(so

r_{2} \in {0, 3, 6}

) and

r_{3}

is even (so

r_{3} \in {0, 2, 4, 6}

), subject to

r_{1} = 6 - r_{2} - r_{3} \geq 0

. The valid triples

(r_{1}, r_{2}, r_{3})

and their contributions are:

(r_{1}, r_{2}, r_{3}) (6, 0, 0) (4, 0, 2) (2, 0, 4) (0, 0, 6) (3, 3, 0) (1, 3, 2) (0, 6, 0) term 1 \frac{6 !}{4 ! 2 !} \cdot 3 = 45 \frac{6 !}{2 ! 4 !} \cdot 9 = 135 \frac{6 !}{6 !} \cdot 27 = 27 \frac{6 !}{3 ! 3 !} \cdot 2 = 40 \frac{6 !}{1 ! 3 ! 2 !} \cdot 2 \cdot 3 = 360 \frac{6 !}{6 !} \cdot 4 = 4

Summing:

1 + 45 + 135 + 27 + 40 + 360 + 4 = 612

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Integer answer