If the Coefficient of \(x^{30}\) in the expansion of \(\left(1+\frac{1}{x}\right)^{6}\left(1+x^{2}\right)^{7}\left(1-x^{3}\right)^{8} ; x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.

Binomial Theorem — JEE Main · 2024 · 01 Feb 2024 · Shift 1 Question with Solution

If the Coefficient of

x^{30}

in the expansion of

(1 + \frac{1}{x})^{6} (1 + x^{2})^{7} (1 - x^{3})^{8}; x \neq = 0

α

, then

∣ α ∣

equals ____.

Answer: 678

Need the coefficient of

x^{30}

(1 + \frac{1}{x})^{6} (1 + x^{2})^{7} (1 - x^{3})^{8} = \frac{( x + 1 ) ^{6} ( 1 + x ^{2} ) ^{7} ( 1 - x ^{3} ) ^{8}}{x ^{6}}

, i.e. the coefficient of

x^{36}

(1 + x)^{6} (1 + x^{2})^{7} (1 - x^{3})^{8}

.
General term: pick

x^{r_{1}}

from

(1 + x)^{6}

(coefficient

(r _{1} 6)

x^{2 r_{2}}

from

(1 + x^{2})^{7}

(coefficient

(r _{2} 7)

x^{3 r_{3}}

from

(1 - x^{3})^{8}

(coefficient

(- 1)^{r_{3}} (r _{3} 8)

), with

r_{1} + 2 r_{2} + 3 r_{3} = 36

0 \leq r_{1} \leq 6, 0 \leq r_{2} \leq 7, 0 \leq r_{3} \leq 8

.
Since

r_{1} \leq 6

and

2 r_{2} \leq 14

, we need

3 r_{3} \geq 36 - 6 - 14 = 16

, so

r_{3} \geq 6

; check

r_{3} = 6, 7, 8

r_{3} = 8

3 r_{3} = 24

, need

r_{1} + 2 r_{2} = 12

with

r_{1} \leq 6

(r_{1}, r_{2}) = (0, 6), (2, 5), (4, 4), (6, 3)

r_{3} = 7

3 r_{3} = 21

, need

r_{1} + 2 r_{2} = 15

with

r_{1} \leq 6, r_{2} \leq 7

(1, 7), (3, 6), (5, 5)

r_{3} = 6

3 r_{3} = 18

, need

r_{1} + 2 r_{2} = 18

with

r_{1} \leq 6, r_{2} \leq 7

(4, 7), (6, 6)

.

Summing

(r _{1} 6) (r _{2} 7) (- 1)^{r_{3}} (r _{3} 8)

over all these terms (with

(- 1)^{r_{3}} = + 1

for

r_{3} = 8, 6

and

- 1

for

r_{3} = 7

r_{3} = 8

(

+ (8 8) = 1

(0 6) (6 7) + (2 6) (5 7) + (4 6) (4 7) + (6 6) (3 7) = (1) (7) + (15) (21) + (15) (35) + (1) (35) = 7 + 315 + 525 + 35 = 882

r_{3} = 7

(

- (7 8) = - 8

(1 6) (7 7) + (3 6) (6 7) + (5 6) (5 7) = (6) (1) + (20) (7) + (6) (21) = 6 + 140 + 126 = 272

; contributes

- 8 \times 272 = - 2176

r_{3} = 6

(

+ (6 8) = 28

(4 6) (7 7) + (6 6) (6 7) = (15) (1) + (1) (7) = 22

; contributes

28 \times 22 = 616

Total coefficient

= 882 - 2176 + 616 = - 678

. So

α = - 678

and

∣ α ∣ = 678

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Integer answer