Find-the-coefficient-of-x-17-in-the-expansion-of-x-1-x-2-x-3-x-19-

Question Number 5003 by 314159 last updated on 30/Mar/16

F i n d t h e c o e f f i c i e n t o f x^{17} i n t h e e x p a n s i o n o f

(x + 1) (x + 2) (x + 3) \dots (x + 19) .

Commented by 123456 last updated on 30/Mar/16

1 + 1 + \dots + 1 (17)

1 + 1 + \dots + 1 (17) + 0

1 + 1 + \dots + 1 (17) + 0 + 0

\frac{19!}{17! (19 - 17)!} = \frac{19 \cdot 18}{6} = 19 \cdot 3 = 57

(57 + 1) + \dots + (57 + 19) = 1083 + 190 = 1273

(57 + 1 + 2) + (57 + 1 + 3) \dots + (57 + 18 + 19)

\frac{19!}{2! (19 - 2)!} = 171

Commented by prakash jain last updated on 31/Mar/16

Coefficient of x^{17} can be generated by multiplication

any 2 constant terms and remaining x terms .

= 1 \cdot 2 + 1 \cdot 3 + \dots + 1 \cdot 19

+ 2.3 + 2.4 + . . + 2.19

+

\dots

+ 18 \cdot 19

= \underset{i = 1}{\sum^{18}} i (\underset{j = i + 1}{\sum^{19}} j)

Arithmetic Progression formulas to sum .