how-many-terms-are-there-in-x-2-x-1-11-

Question Number 87933 by john santu last updated on 07/Apr/20

how many terms are there

in {(x^{2} + x + 1)}^{11} ?

Commented by mr W last updated on 07/Apr/20

x^{0} \dots x^{22}

\Rightarrow 23 t e r m s

Commented by mr W last updated on 07/Apr/20

b u t h o w m a n y t e r m s a r e t h e r e i n

{(x^{5} + x^{2} + 1)}^{11} ?

Commented by john santu last updated on 07/Apr/20

56 terms

Commented by john santu last updated on 07/Apr/20

hihihi \dots . .

Commented by mr W last updated on 07/Apr/20

b u t f o r e x a m p l e w e {d o n}^{'} t h a v e x t e r m,

x^{3} t e r m e t c .

Commented by jagoll last updated on 07/Apr/20

\underset{i + j + k ⩽ 11}{\sum^{11}} (\begin{matrix} 11 \\ i, j, k \end{matrix}) {(x^{5})}^{i} {(x^{2})}^{j} {(1)}^{k}

= \underset{i + j + k ⩽ 11}{\sum^{11}} (\begin{matrix} 11 \\ i, j, k \end{matrix}) x^{5 i + 2 j} {(1)}^{k}

i = 0 \Rightarrow j = 0 \land k = 11,

j = 1 \land k = 9, j = 2 \land k = 7

j = 3 \land k = 5, j = 4 \land k = 3

j = 5 \land k = 1

i = 1 \Rightarrow j = 0 \land k = 6, j = 1 \land k = 4

j = 2 \land k = 2, j = 3 \land k = 0

i = 2 \Rightarrow j = 0 \land k = 1

Σ terms = 11

Commented by jagoll last updated on 07/Apr/20

hehehe

Commented by ajfour last updated on 07/Apr/20

I s t i l l t h i n k, m r W S i r s, o b v i o u s

a n s w e r, c o r r e c t .

Commented by jagoll last updated on 08/Apr/20

what answer this question sir ?

Commented by mr W last updated on 08/Apr/20

w h e n e x p a n d e d, {(x^{5} + x^{2} + 1)}^{11} h a s 51

t e r m s .

{(x^{5} + x^{2} + 1)}^{11} = \sum_{i + j + k = 11} (_{i, j, k}^{11}) {(x^{5})}^{i} {(x^{2})}^{j} 1^{k}

= \sum_{i + j + k = 11} (_{i, j, k}^{11}) x^{5 i + 2 j}

i, j, k ⩾ 0

i + j + k = 11 \Rightarrow 0 ⩽ i + j ⩽ 11

0 ⩽ 5 i + 2 j ⩽ 55

b u t f r o m 0 t o 55 t h e r e a r e o n l y 51

n u m b e r s w h i c h i s a m u l t i p l e o f 5

(m a x . 55) o r a m u l t i p l e o f 2 (m a x . 22)

o r t h e s u m o f b o t h .

f o l l o w i n g t e r m s {d o n}^{'} t e x i s t :

x^{1}, x^{3}, x^{48}, x^{51}, x^{53}

b e c a u s e y o u {c a n}^{'} t e x p r e s s t h e n u m b e r s

1, 3, 48, 51, 53 a s 5 i + 2 j w i t h 0 ⩽ i + j ⩽ 11 .

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