Integers lying between 1000 and 10000 such that the sum of all the digits in each of those numbers becomes 30 . ∴ Coefficient of x30 in the expansion of (x1+x2.‌+...+x9)(x0+x1+...+x9)3 ‌=x(‌
1−x9
1−x
)(‌
1−x10
1−x
)3 ‌=x(1−x9)(1−x10)3(1−x)−4 ⇒ Coefficient of x29 in (1−x9)(1−x10)3(1−x)−4‌ =‌(1−x9)(1−3x10+3x20−x30)(1−x)−4 =‌(1−x9−3x10+3x19+3x20−3x29. ‌‌‌−x30+x39)(1−x)−4 Here, (1−x)−4=
∞
∑
k=0
k+‌3C3xk ∴ Coefficient of x29 from 1⋅(1−x)−4 =‌29+3C3=‌32C3=4960 Coefficient of x20 from −x9⋅(1−x)−4 =−‌20+3C3=−‌23C3=−1771 Coefficient of x19 from −3x10⋅(1−x)−4 ‌=−3(‌19+3C3) ‌=−3(1540)=−4620 Coefficient of x10 from 3x19(1−x)−4 (Means form is 3x19
‌
Σ
k+3C
3xk need x29,k‌=10) ‌=310+‌3C3=3(‌13C3)=858 Coefficient of x9 from 3x20⋅(1−x)−4 =3(‌9+3C3)=3(220)=660 Coefficient of x0 from −3x29⋅(1−x)−4 =−3(‌0+3C3)=−3(1)=−3 ∴ Total numbers ‌=4960−1771−4620+858+660−3 ‌=84