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Question Number 139668 by aupo14 last updated on 30/Apr/21

Commented by mr W last updated on 30/Apr/21

$x^{2} c a n o n l y b e f o r m e d f r o m t w o t i m e s$ $(1 + x + x^{3}) . t h e r e f o r e t h e c o e f f i c i e n t$ $o f x^{2} i s C_{2}^{9} = \frac{9 \times 8}{2} = 36 .$
Commented by aupo14 last updated on 30/Apr/21
can you show me how x^2 will be formed just two times? please?
Commented by mr W last updated on 01/May/21

$w e h a v e 9 t i m e s (1 + x + x^{3}) . t o g e t a x^{2}$ $w e n e e d t h e x t e r m i n t w o o f t h e 9$ $(1 + x + x^{3}) s a n d t h e 1 t e r m i n t h e o t h e r$ $7 (1 + x + x^{3}) s . t o s e l e c t 2 f r o m 9$ $(1 + x + x^{3}) s w e h a v e C_{2}^{9} c o m b i n a t i o n s,$ $t h a t m e a n s w e g e t C_{2}^{9} t i m e s x^{2}, i . e .$ $t h e c o e f . o f x^{2} i s C_{2}^{9} .$ $i f t h e q u e s t i o n i s {(1 + 2 x + 3 x^{3})}^{9}, w e$ $w i l l g e t C_{2}^{9} t i m e s {(2 x)}^{2}, t h e n t h e c o e f .$ $o f x^{2} i s 4 \times C_{2}^{9} = 144 .$
	Answered by mr W last updated on 01/May/21

	${(1 + x + x^{3})}^{9} = \sum_{a + b + c = 9} (\begin{matrix} 9 \\ a, b, c \end{matrix}) 1^{a} x^{b} x^{3 c}$ $b + 3 c = 2 \Rightarrow b = 2, c = 0$ $a + b + c = 9 \Rightarrow a = 7$ $c o e f . o f x^{2} i s (\begin{matrix} 9 \\ 7, 2, 0 \end{matrix}) = \frac{9!}{7! 2! 0!} = 36$
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