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Question Number 84998 by jagoll last updated on 18/Mar/20

$$\mathrm{what}\mathrm{is}\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^{29}$$$$\mathrm{in}\mathrm{expression}{(1+{\mathrm{x}}^5+{\mathrm{x}}^7+{\mathrm{x}}^9)}^{29}$$

Answered by mind is power last updated on 18/Mar/20

$$29=5a+7b+9c$$$$a+b+c+d=29$$$$c=1\Rightarrow 20=5a+7b\Rightarrow b=0,a=4$$$$\Rightarrow d=24$$$$c=0$$$$29=5a+7b\Rightarrow a=3,b=2,$$$$d=24$$$$k=(a,b,c)$$$${(1+x^5+x^7+x^9)}^{29}=\sum _{a+b+c=29-d}\frac{29!}{a!.b!.c!}{x}^{5a+7b+9c}$$$$=(\frac{29!}{4!.24!}+\frac{29!}{3!.2!.24!})x^{29}$$

Commented by jagoll last updated on 18/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Commented by jagoll last updated on 18/Mar/20

$$\mathrm{yes}\mathrm{sir}\mathrm{correct}.\mathrm{via}\mathrm{wolframalpha}$$

Commented by mind is power last updated on 18/Mar/20

$$yeahiforget24!$$$$$$

Answered by mr W last updated on 18/Mar/20

$${(1+{\mathrm{x}}^5+{\mathrm{x}}^7+{\mathrm{x}}^9)}^{29}$$$$=\sum _{a+b+c+d=29}\left({}_{a,b,c,d}^{29}\right)1^d{\left(x^5\right)}^a{\left(x^7\right)}^b{\left(x^9\right)}^c$$$$=\sum _{a+b+c+d=29}\left({}_{a,b,c,d}^{29}\right)x^{5a+7b+9c}$$$$with\left({}_{a,b,c,d}^{29}\right)=\frac{29!}{a!b!c!d!}$$$$$$$$example:term{x}^{29}:$$$$5a+7b+9c=29$$$$\Rightarrow a=3,b=2,c=0\Rightarrow d=24$$$$\Rightarrow a=4,b=0,c=1\Rightarrow d=24$$$$thecoefficientof{x}^{29}:$$$$\frac{29!}{3!2!24!}+\frac{29!}{4!24!}=29\times 28\times 27\times 26\times 25(\frac{1}{12}+\frac{1}{24})$$$$=1781325$$$$$$$$example:term{x}^{50}:$$$$5a+7b+9c=50$$$$\Rightarrow a=3,b=5,c=0\Rightarrow d=21$$$$\Rightarrow a=10,b=0,c=0\Rightarrow d=19$$$$\Rightarrow a=1,b=0,c=5\Rightarrow d=23$$$$thecoefficientof{x}^{50}:$$$$\frac{29!}{3!5!21!}+\frac{29!}{10!19!}+\frac{29!}{5!23!}=263240250$$

Commented by mr W last updated on 18/Mar/20

Commented by mr W last updated on 18/Mar/20

$$myresultcoincideswiththatfrom$$$$wolframalpha,seebelow.$$

Commented by jagoll last updated on 19/Mar/20

$$\mathrm{sir}\mathrm{w}.\mathrm{thank}\mathrm{you}.{\mathrm{i}}^{\prime }\mathrm{m}\mathrm{wrong}$$$$\mathrm{operate}\mathrm{my}\mathrm{wolframaplha}.$$

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