Find-the-coefficient-of-1-x-20-in-1-x-x-2-20-2-x-4-in-1-x-x-2-x-3-11-please-help-me-

Question Number 152817 by rexford last updated on 01/Sep/21

F i n d t h e c o e f f i c i e n t o f

1 . x^{20} i n {(1 - x + x^{2})}^{20}

2 . x^{4} i n {(1 + x + x^{2} + x^{3})}_{}^{11}

p l e a s e, h e l p m e

Answered by puissant last updated on 01/Sep/21

2)

{(1 + x + x^{2} + x^{3})}^{11} = {(1 + x^{2})}^{11} {(1 + x)}^{11}

= (1 + 11 x^{2} + C_{11}^{2} x^{4} + \dots + x^{22}) (1 + 11 x + C_{11}^{2} x^{2} + \dots + x^{11})

H e n c e, c o e f f i c i e n t o f x^{4} w i l l b e

1 \times C_{11}^{4} + 11 \times C_{11}^{2} + 1 \times C_{11}^{2}

= C_{11}^{4} + 12 C_{11}^{2}

= 330 + 660

= 990 . .

Answered by mr W last updated on 02/Sep/21

(1)

{(1 - x + x^{2})}^{20} = \sum_{a + b + c = 20} (\begin{matrix} 20 \\ a, b, c \end{matrix}) 1^{a} {(- x)}^{b} {(x^{2})}^{c}

= \sum_{a + b + c = 20} (\begin{matrix} 20 \\ a, b, c \end{matrix}) {(- 1)}^{b} x^{b + 2 c}

b + 2 c = 20 & a + b + c = 20

\Rightarrow c = 10, b = 0, a = 10 \Rightarrow \frac{20!}{10! 0! 10!}

\Rightarrow c = 9, b = 2, a = 9 \Rightarrow \frac{20!}{9! 2! 9!}

\Rightarrow c = 8, b = 4, a = 8 \Rightarrow \frac{20!}{8! 4! 8!}

\Rightarrow c = 7, b = 6, a = 7 \Rightarrow \frac{20!}{7! 6! 7!}

\Rightarrow c = 6, b = 8, a = 6 \Rightarrow \frac{20!}{6! 8! 6!}

\dots

\Rightarrow c = 2, b = 16, a = 2 \Rightarrow \frac{20!}{2! 16! 2!}

\Rightarrow c = 1, b = 18, a = 1 \Rightarrow \frac{20!}{1! 18! 1!}

\Rightarrow c = 0, b = 20, a = 0 \Rightarrow \frac{20!}{0! 20! 0!}

c o e f . o f x^{20} i s

\frac{20!}{{(10!)}^{2} 0!} + \frac{20!}{{(9!)}^{2} 2!} + \frac{20!}{{(8!)}^{2} 4!} + \dots + \frac{20!}{{(1!)}^{2} 18!} + \frac{20!}{{(0!)}^{2} 20!}

= 184 756 + 9 237 808 + 62 355 150 +

+ 133 024 320 + 116 396 280 + 46 558 512 +

+ 8 817 900 + 775 200 + 29 070 + 380 + 1

= 377 379 369

o r

{(1 - x + x^{2})}^{20} = \frac{{(1 + x^{3})}^{20}}{{(1 + x)}^{20}} = \underset{r = 0}{\sum^{20}} C_{r}^{20} x^{3 r} \underset{k = 0}{\sum^{\infty}} C_{19}^{k + 19} {(- 1)}^{k} x^{k}

3 r + k = 20 :

r = 0, k = 20 \Rightarrow C_{0}^{20} C_{19}^{39}

r = 1, k = 17 \Rightarrow - C_{1}^{20} C_{19}^{36}

r = 2, k = 14 \Rightarrow C_{2}^{20} C_{19}^{33}

r = 3, k = 11 \Rightarrow - C_{3}^{20} C_{19}^{30}

\dots

r = 6, k = 2 \Rightarrow C_{6}^{20} C_{19}^{21}

c o e f . o f x^{20} i s

C_{0}^{20} C_{19}^{39} - C_{1}^{20} C_{19}^{36} + C_{2}^{20} C_{19}^{33} - C_{3}^{20} C_{19}^{30} + \dots + C_{6}^{20} C_{19}^{21}

= 377 379 369

o r

{(1 - x + x^{2})}^{20} = \underset{r = 0}{\sum^{20}} C_{r}^{20} x^{r} {(x - 1)}^{r}

= \underset{r = 0}{\sum^{20}} [C_{r}^{20} x^{r} \underset{k = 0}{\sum^{r}} C_{k}^{r} x^{r - k} {(- 1)}^{k}]

= \underset{r = 0}{\sum^{20}} \underset{k = 0}{\sum^{r}} {(- 1)}^{k} C_{r}^{20} C_{k}^{r} x^{2 r - k}

2 r - k = 20 :

k = 0, r = 10 \Rightarrow C_{10}^{20} C_{0}^{10} = \frac{20!}{{(10!)}^{2} 0!}

k = 2, r = 11 \Rightarrow C_{11}^{20} C_{2}^{11} = \frac{20!}{{(9!)}^{2} 2!}

k = 4, r = 12 \Rightarrow C_{12}^{20} C_{4}^{12} = \frac{20!}{{(8!)}^{2} 4!}

\dots

k = 20, r = 20 \Rightarrow C_{20}^{20} C_{20}^{20} = \frac{20!}{{(0!)}^{2} 20!}

Commented by Rankut last updated on 01/Sep/21

Commented by mr W last updated on 01/Sep/21

Commented by Tawa11 last updated on 01/Sep/21

Weldone sir .

Commented by peter frank last updated on 03/Sep/21

GODBLESS you

Answered by mr W last updated on 01/Sep/21

(2)

{(1 + x + x^{2} + x^{3})}^{11} = \frac{{(1 - x^{4})}^{11}}{{(1 - x)}^{11}}

= \underset{r = 0}{\sum^{11}} {(- 1)}^{r} C_{r}^{11} x^{4 r} \underset{k = 0}{\sum^{\infty}} C_{10}^{k + 10} x^{k}

4 r + k = 4 :

r = 0, k = 4 \Rightarrow C_{0}^{11} C_{10}^{14} = 1001

r = 1, k = 0 \Rightarrow - C_{1}^{11} C_{10}^{10} = - 11

c o e f . o f x^{4} i s

1001 - 11 = 990

Answered by mathmax by abdo last updated on 02/Sep/21

{(1 - x + x^{2})}^{20} = {(x^{2} - x + 1)}^{20} = \sum_{n = 0}^{20} C_{20}^{n} {(x^{2} - x)}^{n}

= \sum_{n = 0}^{20} x^{n} C_{20}^{n} {(x - 1)}^{n}

= \sum_{n = 0}^{20} x^{n} C_{20}^{n} (\sum_{k = 0}^{n} C_{n}^{k} x^{k} {(- 1)}^{n - k})

= \sum_{n = 0}^{20} (\sum_{k = 0}^{n} C_{n}^{k} x^{k} {(- 1)}^{n - k}) C_{20}^{n} x^{n}

= \sum_{n = 0}^{20} (\sum_{k = 0}^{n} C_{n}^{k} x^{k} {(- 1)}^{k}) {(- 1)}^{n} C_{20}^{n} x^{n}

= \sum_{n = 0}^{20} (C_{n}^{o} - C_{n}^{1} x + C_{n}^{2} x^{2} + \dots . C_{n}^{n} {(- 1)}^{n} x^{n}) {(- 1)}^{n} C_{20}^{n} x^{n}

= \sum_{n = 0}^{20} C_{n}^{0} {(- 1)}^{n} C_{20}^{n} x^{n} - \sum_{n = 0}^{20} C_{n}^{1} {(- 1)}^{n} C_{20}^{n} x^{n + 1} +

+ \sum_{n = 0}^{20} C_{n}^{2} {(- 1)}^{n} C_{20}^{n} x^{n + 2} + \dots . + \sum_{n = 0}^{20} C_{n}^{n} C_{20}^{n} x^{2 n} \Rightarrow

lecoefficient de x^{n} in this expansion is

A_{n} = C_{20}^{0} C_{20}^{20} + C_{19}^{1} C_{20}^{19} + C_{18}^{2} C_{20}^{18} + \dots . + C_{10}^{10} C_{20}^{10}

\Rightarrow A_{n} = {(C_{20}^{o})}^{2} + {(C_{19}^{1})}^{2} + {(C_{18}^{2})}^{2} + \dots . . {(C_{10}^{10})}^{2}